BRAVE NEW MOTIVIC HOMOTOPY THEORY I: THE …BRAVE NEW MOTIVIC HOMOTOPY THEORY I: THE LOCALIZATION...

BRAVE NEW MOTIVIC HOMOTOPY THEORY I:

THE LOCALIZATION THEOREM

ADEEL A. KHAN

Abstract. This series of papers is dedicated to the study of motivic homotopy theory in thecontext of brave new or spectral algebraic geometry. In Part I we set up the theory and prove

an analogue of the localization theorem of Morel–Voevodsky in classical motivic homotopy

theory.

Contents

1. Introduction 21.1. Main result 21.2. Summary of construction 41.3. Contents 51.4. Conventions 61.5. Acknowledgments 62. Spectral algebraic geometry 62.1. Spectral prestacks 62.2. Quasi-coherent sheaves 72.3. Spectral stacks 82.4. Spectral schemes 82.5. Closed immersions 92.6. Vector bundles 92.7. Infinitesimal extensions 112.8. The cotangent sheaf 122.9. Smooth and etale morphisms 132.10. Deformation along infinitesimal extensions 142.11. Push-outs of closed immersions 142.12. Lifting smooth morphisms along closed immersions 152.13. Regular closed immersions 153. Abstract Morel–Voevodsky homotopy theory 173.1. Presheaves 173.2. Left localizations 173.3. Pointed objects 183.4. Spectrum objects 193.5. Homotopy invariance 223.6. Excision structures 223.7. Topological excision structures 233.8. Unstable homotopy theory 253.9. Pointed homotopy theory 263.10. Stable homotopy theory 274. Brave new motivic homotopy theory 294.1. Nisnevich excision 29

Date: June 29, 2017.

1

2 ADEEL A. KHAN

4.2. Homotopy invariance 314.3. Motivic spaces 324.4. Pointed motivic spaces 334.5. Motivic spectra 345. Inverse and direct image functoriality 345.1. For motivic spaces 345.2. For pointed motivic spaces 355.3. For motivic spectra 366. Functoriality along smooth morphisms 366.1. The functor p] 366.2. Smooth base change formulas 386.3. Smooth projection formulas 397. Functoriality along closed immersions 427.1. The exceptional inverse image functor i! 427.2. The localization theorem 427.3. A reformulation of localization 457.4. Nilpotent invariance 457.5. Closed base change formula 457.6. Closed projection formula 467.7. Smooth-closed base change formula 478. The localization theorem 488.1. The space of Z-trivialized maps 488.2. Etale base change 508.3. Reduction to the case of vector bundles 528.4. Motivic contractibility of hS(X, t) 538.5. Proof of the localization theorem 54Appendix A. Coefficient systems 55A.1. Passage to right/left adjoints 55A.2. Adjointable squares 56A.3. Coefficient systems 57A.4. Left-adjointability 58A.5. Right-adjointability 60A.6. Biadjointability 61Appendix B. Topologically quasi-cocontinuous functors 62B.1. Reduced presheaves 62B.2. Topologically quasi-cocontinuous functors 63References 65

1. Introduction

1.1. Main result.

1.1.1. Given a scheme S, let H(S) denote the ∞-category of motivic spaces over S.

One of the most fundamental results in motivic homotopy theory is the localization theoremof F. Morel and V. Voevodsky, one formulation of which is as follows. Let i : Z → S be a closedimmersion with quasi-compact open complement j : U → S, and consider the direct imagefunctor i∗ : H(Z) → H(S). The localization theorem states that this functor is fully faithful,with essential image spanned by motivic spaces F whose restriction j∗(F) is contractible.

BRAVE NEW MOTIVIC HOMOTOPY THEORY I: THE LOCALIZATION THEOREM 3

The main goal of this paper is to prove the analogue of this theorem in spectral algebraicgeometry.

1.1.2. Before proceeding, let us explain why we chose this theorem as the starting point of ourstudy of motivic homotopy theory in spectral algebraic geometry.

Every motivic spectrum E over S represents a generalized motivic cohomology theoryHq(−,E(p)) on S-schemes (p, q ∈ Z), like motivic cohomology, homotopy invariant algebraicK-theory, or algebraic cobordism.

The localization theorem gives rise to the expected long exact localization sequences inE-cohomology, for any motivic spectrum E:

· · · → HqZ(S,E(p))

i!−→ Hq(S,E(p))j∗−→ Hq(U,E(p))

∂−→ Hq+1Z (S,E(p))→ · · · ,

for any closed immersion i : Z → S with open complement j : U → S. Here HqZ(S,E(p))

denotes1 E-cohomology of S with support in Z.

The localization theorem is also a crucial input into J. Ayoub’s construction of the formalismof six operations in motivic homotopy theory. The construction of the six operations in thesetting of spectral schemes will be the subject of one of the sequels of this paper, and thelocalization theorem will play an equally important role.

1.1.3. Let us also say a few words about why we are interested in developing motivic homotopytheory in the setting of spectral algebraic geometry.

In this setting, the localization theorem has another very interesting consequence which wedid not mention yet. Recall that a spectral scheme S is, roughly speaking2, a pair (Scl,OS), whereScl is a classical scheme and OS is a sheaf of connective E∞-ring spectra whose underlying sheafof commutative rings π0(OS) coincides with the structure sheaf OScl

. The underlying classicalscheme Scl may be viewed as a spectral scheme with discrete structure sheaf, and there is acanonical map Scl → S which is a closed immersion in the sense of spectral algebraic geometry.Since its open complement is empty, the localization theorem will have the consequence thatthe motivic homotopy category over S is equivalent to the motivic homotopy category over Scl.We refer to this phenomenon as “derived nilpotent invariance” (we think of sections of πi(OS),for i > 0, as derived nilpotents).

We should note that it will not be obvious from our construction that, over the classicalscheme Scl, the motivic homotopy category we construct is equivalent to the usual motivichomotopy category. This turns out to be nevertheless true, and it will also be a consequence ofthe localization theorem, but its proof will be postponed to Part II [CK17] of this series.

In other words, any generalized motivic cohomology theory can be extended to spectralalgebraic geometry in a trivial way:

Hq(X,E(p)) := Hq(Xcl,E(p)),

and the claim is that every generalized motivic cohomology theory in spectral algebraic geometryarises in this way.

Let us mention one interesting application. Let X be an arbitrary scheme. Given any regularclosed immersion i : Z → X, where we allow Z to be an arbitrary spectral scheme3, one can

1In the case where Z and S are both smooth over some base, i is pure of codimension d, and the spectrum Eis oriented, Morel–Voevodsky’s relative purity theorem will further identify this term with Hq−2d(Z,E(p− d)).

2See [Lur16b, Def. 1.1.2.8] for a precise version of this definition; we will give an alternative definition in

Sect. 2.3See Definition 2.13.2 for our definition of regular closed immersion in this context.

4 ADEEL A. KHAN

construct an associated (virtual) fundamental class [Z] such that the formula

[Z1] · [Z2] = [Z1×X

Z2]

always holds, where Z1 and Z2 are regularly immersed closed spectral subschemes of X, and thefibred product Z1×X Z2 is taken in the category of spectral schemes. This formula holds in anygeneralized motivic cohomology theory (in the above-mentioned sense) and can be viewed as avast generalization of Bezout’s theorem4. We will discuss this in detail in a sequel to this paper.

1.2. Summary of construction. Below we briefly describe our construction of the motivichomotopy category over a spectral scheme.

1.2.1. Let S be a spectral scheme (we will always assume that S is quasi-compact and quasi-

separated). A motivic space over S will be a presheaf of spaces F on the ∞-category SmE∞/S of

smooth spectral S-schemes (of finite presentation), satisfying the following properties:

(Red) The space Γ(∅,F) is contractible.

(Nis) Let Q be a Nisnevich square, i.e. a cartesian square of smooth spectral S-schemes

(1.1)

W V

U X

p

j

where j is an open immersion, p is etale, and the induced morphism p−1(X−U)→ X−U is anisomorphism of reduced classical schemes. Then the induced square of spaces

Γ(X,F) Γ(U,F)

Γ(V,F) Γ(W,F)

j∗

p∗

is homotopy cartesian.

(Htp) Let A1 denote the brave new affine line, i.e. the affine spectral scheme Spec(St), whereS is the sphere spectrum, and St denotes the free E∞-algebra on one generator t (in degreezero). Then for every smooth spectral S-scheme, the canonical map

Γ(X,F)→ Γ(X×A1,F)

is invertible.

We write HE∞(S) for the ∞-category of motivic spaces over S. This is a left localization of

the∞-category of presheaves of spaces on SmE∞/S , and we write F 7→ Lmot(F) for the localization

functor.

Any smooth spectral S-scheme X represents a presheaf of spaces hS(X), whose localizationwe denote by MS(X) := Lmot hS(X).

1.2.2. The assignment S 7→ HE∞(S) admits the following functorialities.

Given a morphism f : T→ S of spectral schemes, there is an inverse image functor

f∗H : HE∞(S)→ HE∞(T)

which is given on representables by the assignment MS(X) 7→ MT(X×S T). It is left adjoint to adirect image functor

fH∗ : HE∞(T)→ HE∞(S).

4For X a smooth projective variety over the field of complex numbers, a similar formula taking values in

singular cohomology was announced by J. Lurie [Lur16b].


We also have the operations (⊗,Hom), where the bifunctor (F,G) 7→ F ⊗ G which is given onrepresentables by MS(X)⊗MS(Y) = MS(X×S Y).

If f is smooth, there is an “extra” operation

fH] : HE∞(T)→ HE∞(S),

left adjoint to f∗, which is given on representables by the assignment MT(X) 7→ MS(X). Notethat this operation is not one of the “six operations”, but it is compatible with the four operations(f∗, f∗,⊗,Hom) in the sense that it satisfies various base change and projection formulas, whichare the subject of Sect. 6.

1.2.3. One consequence of the localization theorem is that there are well-defined operations(i!, i

!) (called exceptional direct and inverse image, respectively) for any closed immersion i,satisfying base change and projection formulas (see Sect. 7). In order to extend these operationsto more general morphisms, it is necessary to pass to the stable version of motivic homotopytheory.

Classically, stable motivic homotopy theory over S is obtained from the unstable versionby formally inverting the Thom spaces of all vector bundles over S, with respect to the smashproduct. Because of descent, it suffices in fact to invert the Thom space of the affine line A1

S,which is identified up to A1-homotopy with the motivic space represented by the projective lineP1

S, pointed by the section at infinity. The latter can be further identified up to A1-homotopywith the S1-suspension of the motivic space represented by the Gm,S, pointed by the unit section.Thus the process of stabilization can be done in two steps, first taking the S1-stabilization andthen Gm-stabilization.

This whole discussion remains valid in the setting of spectral algebraic geometry. Howeverwe will defer our discussion of the P1-stable theory to Part III, since the localization theoremalready holds in the unstable category.

1.3. Contents. In Sect. 2 we review the basics of spectral algebraic geometry. We follow the“functor of points” approach as in [TV08] or [AG14]; an alternative approach using locally ringed∞-toposes can be found in [Lur16b]. Most of the material is well-known, so we generally omitproofs, except in the last few paragraphs where we include some geometric lemmas that areused in the proof of our main result.

Sect. 3 is an axiomatic exposition of Morel–Voevodsky homotopy theory in general settings.The framework encompasses even equivariant motivic homotopy theory [Hoy17] and noncommu-tative motivic homotopy theory [Rob15]. That said, there is no original mathematics in thissection; it only serves to collect in one place some basic results which we will be referring torepeatedly elsewhere in the paper.

In Sect. 4 we specialize to give the construction of the brave new motivic homotopy categoryover a spectral scheme. We also define its pointed and S1-stable variants. We defer discussion ofthe P1-stable variant to Part III.

In Sect. 5 we discuss the standard functorialities (f∗, f∗) that the assignment S 7→ HE∞(S) isequipped with. We show in Sect. 6 that for any smooth morphism p, the functor p∗ admits aleft adjoint p]. We show that this is compatible with the operations f∗ and ⊗ in the sense thatit satisfies suitable base change and projection formulas.

Sect. 7 deals with the functor i∗ of direct image along a closed immersion. Our first resulthere is that, in the pointed setting, it admits a right adjoint i!. Next we state the main result ofthis paper, the localization theorem (Theorem 7.2.6), which says that i∗ is fully faithful, andidentifies its essential image. We then review some consequences, including nilpotent invariance,as well as some compatibilities between the operation i∗ and the operations f∗, p], and ⊗.

6 ADEEL A. KHAN

Sect. 8 is dedicated to the proof of the localization theorem. Our proof follows the same generaloutline used by Morel–Voevodsky [MV99] in the setting of classical base schemes. Followingsome ideas introduced by M. Hoyois [Hoy17] in the equivariant setting, we make the proof robustenough that it survives in the setting of spectral schemes, and without unnecessary noetherianassumptions made in the original proof of Morel–Voevodsky. Thus our proof, if interpretedin the setting of classical algebraic geometry, gives a proof of Morel–Voevodsky’s theorem forclassical schemes, over arbitrary quasi-compact quasi-separated base schemes.

Appendix A introduces an axiomatic framework which we find convenient for discussing basechange and projection formulas in abstract categories of coefficients.

Appendix B deals with some technical but very elementary topos theory that is used in theproof that the direct image functor i∗ commutes with contractible colimits.

1.4. Conventions. We will use the language of ∞-categories freely throughout the text. Theterm “category” will mean “∞-category” (= quasi-category) by default. Though we will use thelanguage in a model-independent way, we fix for concreteness the model of quasi-categories asdeveloped by A. Joyal and J. Lurie. Our main references are [Lur09] and [Lur16a].

Starting from § 4, all spectral schemes (and classical schemes) will be quasi-compact quasi-separated. Smooth morphisms will always be assumed to be of finite presentation.

1.5. Acknowledgments. The author would like to thank Benjamin Antieau, Denis-CharlesCisinski, Marc Hoyois, and Marc Levine for many helpful discussions, encouragement, andfeedback on previous versions of this paper.

2. Spectral algebraic geometry

In this section we review the theory of spectral algebraic geometry. All the material coveredis standard and can be found in [Lur16b] and [TV08], with the exception of Proposition 2.12.2.

2.1. Spectral prestacks.

2.1.1. Let E∞-Alg(Spt)>0 denote the category of connective E∞-ring spectra.

The Eilenberg–MacLane functor defines a fully faithful functor CAlg → E∞-Alg(Spt)>0,whose essential image is spanned by the discrete (= 0-truncated) connective E∞-ring spectra.

2.1.2. A spectral prestack is a presheaf of spaces on the category E∞-Alg(Spt)op>0, i.e. a functor

E∞-Alg(Spt)>0 → Spc.

We write PreStk for the category of spectral prestacks.

2.1.3. Given a connective E∞-ring spectrum A, we will write Spec(A) for the spectral prestackrepresented by A. We say that a spectral prestack is an affine spectral scheme if it is representedby a connective E∞-ring spectrum, and write Schaff for the full subcategory of PreStk spannedby affine spectral schemes.

Let S be a spectral prestack. For a connective E∞-ring spectrum A, we say that an A-pointof S is a morphism s : Spec(A)→ S, or equivalently a point of the space S(A).


2.1.4. A classical5 prestack is a presheaf on the opposite of the ordinary category CAlg ofcommutative rings.

Given a spectral prestack S, let Scl denote its underlying classical prestack, defined as therestriction to ordinary commutative rings. The functor S 7→ Scl admits a fully faithful leftadjoint, embedding the category of classical prestacks as a full subcategory of spectral prestacks.

We refer to spectral prestacks of the form Spec(A), with A an ordinary commutative ring, asclassical affine schemes. The functor S 7→ Scl sends spectral affine schemes to classical affinespectral schemes: we have Spec(A)cl = Spec(π0(A)) for any connective E∞-ring spectrum A.

2.2. Quasi-coherent sheaves.

2.2.1. Let S = Spec(A) be an affine spectral scheme. A quasi-coherent module on S is thedatum of an A-module. We write OSpec(A) for the quasi-coherent module given by A, viewed asa module over itself.

2.2.2. Let S be a spectral prestack. A quasi-coherent OS-module consists of the following data:

(1) For every affine spectral scheme Spec(A) and every morphism s : Spec(A) → S, aquasi-coherent module Fs on OSpec(A).

(2) For every pair of morphisms s : Spec(A)→ S, s′ : Spec(B)→ S fitting into a commutativetriangle

Spec(A) S,

Spec(B)

s

fs′

an isomorphism f∗(Fs′)∼−→ Fs.

(3) A homotopy coherent system of compatibilities between all such isomorphisms.

2.2.3. More precisely, we define the category Qcoh(S) as the limit

Qcoh(S) := lim←−Spec(A)→S

Qcoh(Spec(A))

in the category of presentable ∞-categories.

This category is stable, as the property of stability is stable under limits of ∞-categories.

Better yet, we can define a presheaf of symmetric monoidal presentable ∞-categories S 7→Qcoh(S) as the right Kan extension of the presheaf A 7→ A-mod along the Yoneda embeddingE∞-Alg(Spt)op

>0 → PreStk.

In particular, for each morphism of spectral prestacks f , we have a symmetric monoidalcolimit-preserving functor f∗, the inverse image functor, and its right adjoint f∗, the directimage functor.

5The adjective classical refers to the fact that they are defined on non-derived objects (ordinary commutativerings, not E∞-ring spectra). In the literature they have been studied by C. Simpson and others under the namehigher prestacks, since they may take values in arbitrary spaces, not just groupoids. In our terminology, prestacksare “higher” by default, and “non-higher” prestacks are 1-truncated prestacks.

8 ADEEL A. KHAN

2.2.4. Let S be a spectral prestack. We write OS for the quasi-coherent module defined byOS,s = OSpec(A) for each affine spectral scheme Spec(A) and each A-point s : Spec(A)→ S. Thisis the unit of the symmetric monoidal structure.

Given a quasi-coherent module F on S, we write Γ(X,F) for the space of sections over aspectral S-scheme X. This is by definition the mapping space Maps(OX, p

∗(F)), where p : X→ Sis the structural morphism.

2.3. Spectral stacks.

2.3.1. Let f : T→ S be a morphism of affine spectral schemes, S = Spec(A), T = Spec(B).

Definition 2.3.2.

(i) The morphism f is locally of finite presentation if B is compact in the category of E∞-A-algebras.

(ii) The morphism f is flat if the underlying A-module B is a filtered colimit of finitely generatedfree A-modules. Equivalently, the morphism fcl : Spec(π0(B)) → Spec(π0(A)) of underlyingclassical schemes is flat (i.e. π0(B) is flat in the usual sense as an π0(A)-module), and thecanonical morphism πi(A)⊗π0(A) π0(B)→ πi(B) is invertible for each i.

(iii) The morphism f is an open immersion if it is locally of finite presentation, flat, and amonomorphism6. Equivalently, it is flat and the morphism fcl : Spec(π0(B))→ Spec(π0(A)) ofunderlying classical schemes is an open immersion (in the classical sense).

Remark 2.3.3. We will use the following observation repeatedly: let f : T→ S be a morphismof affine spectral schemes; if f is flat and S is classical, then T is also classical. Similarly, iff : T→ S is a flat morphism of affine spectral schemes, then the commutative square

Tcl T

Scl S

fcl f

is cartesian.

2.3.4. The Zariski topology on Schaff is the Grothendieck topology associated to the followingpretopology. A family of morphisms of affine spectral schemes (jα : Uα → X)α∈Λ is Zariskicovering if and only if each jα is an open immersion, and the family of functors (jα)∗ : Qcoh(X)→Qcoh(Uα) is conservative.

Definition 2.3.5. A spectral stack is a spectral prestack satisfying descent with respect to theZariski topology.

2.4. Spectral schemes.

2.4.1. Let j : U→ S be a morphism of spectral stacks.

Definition 2.4.2.

(i) If S is affine, then j is an open immersion if it is a monomorphism, and there exists a family(jα : Uα → S)α, with each jα an open immersion of affine spectral schemes Definition 2.3.2 thatfactors through U and induces an effective epimorphism tαUα → U.

(ii) For general S, the morphism j is an open immersion if for each connective E∞-ring spectrumA and each A-point s : Spec(A) → S, the base change U×S Spec(A) → Spec(A) is an openimmersion in the sense of (i).

6I.e. the canonical morphism T→ T×S T is invertible.


2.4.3. Let S be a spectral stack. A Zariski cover of S is a small family of open immersionsof spectral stacks (jα : Uα → S)α such that the canonical morphism tαUα → S is surjective(i.e. an effective epimorphism in the topos of spectral stacks). If each Uα is an affine spectralscheme, we call this an affine Zariski cover.

We define:

Definition 2.4.4. A spectral scheme is a spectral stack S which admits an affine Zariski cover.

We write Sch for the category of spectral schemes, a full subcategory of the category of stacks.It admits coproducts and fibred products.

Definition 2.4.5.

(i) A spectral scheme S is quasi-compact if for any Zariski cover (jα : Uα → S)α∈Λ, there existsa finite subset Λ0 ⊂ Λ such that the family (jα)α∈Λ0

is still a Zariski cover.

(ii) A morphism of spectral schemes f : T→ S is quasi-compact if for any connective E∞-ringspectrum A and any A-point s : Spec(A)→ S, the spectral scheme T×S Spec(A) is quasi-compact.

(iii) A spectral scheme S is quasi-separated if for any open immersions U → S and V → S, withU and V affine, the intersection U×S V is quasi-compact.

2.4.6. We define a classical scheme to be a Zariski sheaf of sets on the category (CAlg)op,admitting a Zariski affine cover. This is equivalent to the definition of scheme given in [GD71].

Given a spectral scheme S, the underlying classical prestack Scl takes values in sets, and is aclassical scheme. We therefore refer to Scl as the underlying classical scheme of S.

2.5. Closed immersions.

2.5.1. Let f : Y → X be a morphism of spectral schemes. We say that the morphism f isaffine if, for any connective E∞-ring spectrum A and A-point x : Spec(A)→ X, the base changeY×X Spec(A) is an affine spectral scheme.

2.5.2. If X = Spec(A) and Y = Spec(B) are affine, a morphism i : Y → X is a closed immersionif the homomorphism A→ B induces a surjection π0(A)→ π0(B).

In general a morphism i : Y → X is a closed immersion if it is affine, and for any connectiveE∞-ring spectrum A and A-point x : Spec(A)→ X, the base change Y×X Spec(A)→ Spec(A)is a closed immersion of affine spectral schemes.

Equivalently, i is a closed immersion if and only if it induces a closed immersion on underlyingclassical schemes.

2.5.3. A nil-immersion is a closed immersion i : Y → X which induces an isomorphismYred → Xred on underlying reduced classical schemes.

2.5.4. Let i : Z → S be a closed immersion of spectral schemes. Let U be the spectralprestack defined as follows: for a connective E∞-ring spectrum A, its A-points are A-pointss : Spec(A)→ S such that the base change Spec(A)×S Z is the empty spectral scheme. One canshow that U is a spectral scheme, and that the canonical morphism U→ S is an open immersion.

We call j : U → S the complementary open immersion to i.

2.6. Vector bundles.

10 ADEEL A. KHAN

2.6.1. Just as in Paragraph 2.2, we can define a notion of quasi-coherent algebra on a spectralprestack S, such that the category of quasi-coherent algebras on Spec(A) coincides with thecategory of A-algebras.

2.6.2. Let S be a spectral scheme and A a quasi-coherent algebra on S. Consider the presheafSpecS(A) on the category of spectral schemes over S, which sends a spectral S-scheme X withstructural morphism f to the space of quasi-coherent algebra homomorphisms Maps(f∗(A),OX).

The presheaf SpecS(A) clearly satisfies Zariski descent. Hence it defines a spectral stack overS (there is a canonical equivalence Sh(Sch/S) = Sh(Sch)/S = Sh(Schaff)/S), which we call therelative spectrum of the quasi-coherent algebra A.

Further, we have:

Lemma 2.6.3. Let A be a connective quasi-coherent algebra over a spectral scheme S. Thenthe spectral stack SpecS(A) is a spectral scheme.

This follows from functoriality in S, and the fact that for S = Spec(A) affine, we haveSpecS(A) = Spec(Γ(S,A)).

2.6.4. For a connective E∞-ring spectrum A and an A-module M, we write SymA(M) for thefree A-algebra generated by M. The assignment M 7→ SymA(M) defines a functor, left adjointto the forgetful functor from A-algebras to A-modules, so that there are canonical isomorphisms

MapsA-alg(SymA(M),B)∼−→ MapsA-mod(M,B)

bifunctorial in M and B.

When M is free of rank n, we will also use the notation

At1, . . . , tn := SymA(A⊕n),

following [Lur16b].

2.6.5. Let F be a quasi-coherent module on S. The quasi-coherent algebra SymOS(F) is

defined by SymOS(F)s := SymOS,s

(Fs) for each connective E∞-ring spectrum A and A-point

s : Spec(A)→ S.

2.6.6. A connective quasi-coherent module F on S is locally free of rank n if there exists a Zariskicover (jα : Uα → S)α such that each inverse image j∗α(F) is a free quasi-coherent OSα -module ofrank n, i.e. j∗α(F) ≈ O⊕nSα

.

Given a locally free module E of finite rank, we define:

Definition 2.6.7. The vector bundle associated to E is the spectral S-scheme E = SpecS(SymOS(E)).

Note that any global section s ∈ Γ(S,E) defines a section s : S → E of the structuralmorphism, which is a closed immersion. In particular, every vector bundle admits a zero section.

2.6.8. For an integer n > 0, we define the affine space of dimension n over a spectral scheme S,to be the total space of the free OS-module O⊕nS :

AnS := SpecS(SymOS

(O⊕nS )).

We will write simply An := AnS for the affine spaces over the sphere spectrum S. Thus

An ≈ Spec(St1, . . . , tn).

2.6.9. For any morphism of spectral schemes f : T → S, we have canonical isomorphismsAn

S ×S T ≈ AnT.


2.6.10. The affine line A1S over S is the affine space of dimension 1. Since the quasi-coherent

module OS has a unit section (being a quasi-coherent algebra), the affine line admits both azero and a unit section.

2.6.11. The underlying classical scheme of AnS is the classical affine space over Scl. We note

however that, even when S is classical, the spectral scheme AnS will not be classical, except in

characteristic zero. We will look at another version of affine space in [CK17, Para. 4.2] that isflat, and does agree with the classical affine space.

2.7. Infinitesimal extensions.

2.7.1. Let p : Y → X be a morphism of affine spectral schemes, X = Spec(A), Y = Spec(B).Given a connective quasi-coherent module F on Y, we let

Y → YF := Spec(B⊕M)

denote the trivial infinitesimal extension of Y along F, where M = Γ(Y,F).

The morphism Y → YF is the closed immersion induced by the homomorphism B⊕M→ B,(b,m) 7→ b.

2.7.2. A derivation of Y over X with values in F, is a retraction of the morphism Y → YF

(in the category of affine spectral schemes over X). There is a canonical retraction, the trivialderivation dtriv, defined by the morphism B→ B⊕M, b 7→ (b, 0).

Let Der(Y/X,F) denote the space of derivations in F.

2.7.3. Let F be a 0-connected quasi-coherent module on Y.

Any derivation d of Y/X valued in F gives rise to an infinitesimal extension i : Y → Yd.This is the closed immersion (in fact, nil-immersion7) defined as the cobase change of the trivialderivation along d, so that there is a cocartesian square

(2.1)

YF Y

Y Yd

dtriv

d

i

in the category of affine spectral schemes.

2.7.4. The following important fact often allows argument by induction along infinitesimalextensions (see e.g. [Lur16a, Prop. 7.1.3.19]).

Proposition 2.7.5. Let S = Spec(A) be an affine spectral scheme. Then there exists a sequenceof nil-immersions of affine spectral schemes

(2.2) Scl = S60 → S61 → · · · → S6n → · · · → S,

with S6n = Spec(A6n), satisfying the following properties:

(i) For each n > 0, the homomorphism A → A6n identifies A6n as the n-truncation of theconnective E∞-ring spectrum A.

(ii) The sequence is functorial in A.

(iii) The canonical morphism A→ lim←−n>0A6n is invertible.

(iv) Each morphism S6n → S6n+1 (n > 0) is an infinitesimal extension by a derivation valuedin πn(OS)[n+ 1].

7See (2.5.3).

12 ADEEL A. KHAN

Further, this sequence is uniquely characterized, up to isomorphism of diagrams indexed onthe poset of nonnegative integers, by the property (i).

The sequence (2.2) is called the Postnikov tower of A.

2.8. The cotangent sheaf.

2.8.1. Let p : Y → X be a morphism of affine spectral schemes. We have:

Proposition 2.8.2. The functor F 7→ Der(Y/X,F) is representable by a connective quasi-coherent module T∗Y/X := T∗p on Y.

The quasi-coherent module T∗Y/X on Y is called the (relative) cotangent sheaf of the morphism

p : Y → X. We obtain the absolute cotangent sheaf T∗S by taking the relative cotangent sheaf ofthe unique morphism S→ Spec(S).

2.8.3. The following lemma describes a transitivity property of the relative cotangent sheaf:

Lemma 2.8.4. Let Zg−→ Y

f−→ X be a sequence of morphisms of affine spectral schemes. Thenthere is a canonical exact triangle

(2.3) g∗(T∗Y/X)→ T∗Z/X → T∗Z/Y

of quasi-coherent sheaves on Z.

In particular, we see that the relative cotangent sheaf T∗Y/X is the cofibre of the canonical

morphism f∗(T∗X)→ T∗Y.

2.8.5. The cotangent sheaf of a vector bundle has a particularly simple description:

Lemma 2.8.6. Let S be an affine spectral scheme. For any connective quasi-coherent moduleF on S, let E denote the spectral S-scheme SpecS(SymOS

(F)). Then we have a canonicalisomorphism

T∗E/S∼−→ p∗(F),

where p denotes the structural morphism E→ S.

2.8.7. Let p : Y → X be a morphism of spectral schemes. For any connective E∞-ring spectrumA, A-point y : Spec(A)→ Y, and 0-connected quasi-coherent module F on Y, a derivation at yof p with values in F is a commutative triangle

Spec(A)

Spec(A)F Y

y

d

in the category of spectral X-schemes.

We write Dery(Y/X,F) for the space of derivations at y.

2.8.8. The functor F 7→ Dery(Y/X,F) is represented by a connective quasi-coherent moduleT∗Y/X,y on Spec(A), called the relative cotangent sheaf of p at y:

Dery(Y/X,F)∼−→ MapsQcoh(Spec(A))(T

∗Y/X,y,F).


2.8.9. Given a connective E∞-ring spectrum A and an A-point y : Spec(A)→ Y, a connectiveE∞-ring spectrum B and a B-point y′ : Spec(B)→ Y, and a morphism of affine spectral schemesf : Spec(B) → Spec(A) with an identification y f ≈ y′, we obtain a canonical morphism ofquasi-coherent modules on Spec(B)

f∗(T∗Y/X,y)→ T∗Y/X,y′

which is invertible.

Moreover, the data of the quasi-coherent modules T∗Y/X,y, as y varies over A-points of Y

(with A an arbitrary connective E∞-ring spectrum), together with the above isomorphisms, iscompatible in a homotopy coherent way, and can therefore be refined to a connective quasi-coherent module T∗Y/X defined on the spectral scheme Y.

2.9. Smooth and etale morphisms. In this paragraph we review some standard materialon smooth and etale morphisms in spectral algebraic geometry from [TV08], [Lur16a], and[Lur16b].

2.9.1. Let p : Y → X be a morphism of affine spectral schemes, with X = Spec(A) andY = Spec(B). We will write T∗Y/X for the relative cotangent sheaf, i.e. the connective quasi-

coherent OY-module corresponding to the connective B-module LB/A (= the cotangent complex).

We define:

Definition 2.9.2.

(i) The morphism p is etale if it is locally of finite presentation and the cotangent sheaf T∗Y/X iszero.

(ii) The morphism p is smooth if it is locally of finite presentation and the cotangent sheaf T∗Y/Xis locally free of finite rank.

Remark 2.9.3. Our use of the term smooth corresponds to differentially smooth in the sense of[Lur16b, Def. 11.2.2.2].

We have:

Lemma 2.9.4.

(i) The set of etale (resp. smooth) morphisms is stable under composition and base change.

(ii) Open immersions are etale, and etale morphisms are smooth.

(iii) A morphism p : Y → X is etale if and only if it is flat and the underlying morphism ofclassical schemes pcl : Ycl → Xcl is etale in the sense of [Gro67].

Remark 2.9.5. In [CK17, Para. 4.1] we will consider an alternative notion of smoothness calledfibre-smoothness, which amounts to being flat, and smooth on underlying classical schemes. Itis important to note that smoothness is not equivalent to fibre-smoothness. For example, themorphism HFp → HFp[t] has the latter property, but is not smooth (see [TV08, Prop. 2.4.1.5]).

2.9.6. We now extend the above definitions to morphisms of spectral schemes, by defining themZariski-locally on the source. That is:

Definition 2.9.7. A morphism of spectral schemes p : Y → X is smooth (resp. etale, flat, locallyof finite presentation) if there exist affine Zariski covers (Yα → Y)α and (Xβ → X)β togetherwith the data of, for each α, an index β and a morphism of affine spectral schemes Yα → Xβ

14 ADEEL A. KHAN

which is smooth (resp. etale, flat, locally of finite presentation) and fits in a commutative square

Yα Xβ

Y X.

Example 2.9.8. For any vector bundle E over a spectral scheme S, the projection π : E→ S issmooth. In particular, the affine space An

S is smooth over S for each n > 0.

2.9.9. The following is a brave new version of [Gro67, Thm. 17.11.4] (see [Lur16a, Prop.11.2.2.1]).

Proposition 2.9.10. A morphism p : Y → X is smooth if and only if, Zariski-locally on Y,there exists a factorization of p as a composite

(2.4) Yq−→ X×An r−→ X

for some integer n > 0, where q is etale and r is the canonical projection.

This implies (see [Lur16a, Props. 11.2.4.1 and 11.2.4.3]):

Lemma 2.9.11. If a morphism p : Y → X is smooth, then the induced morphism pcl : Ycl → Xcl

of underlying classical schemes is smooth.

2.10. Deformation along infinitesimal extensions.

2.10.1. Let S be an affine spectral scheme and S′ the infinitesimal extension of S by a derivationd : T∗S → F, for some 0-connected quasi-coherent module F. Let X be an affine spectral schemeover S with structural morphism p.

Definition 2.10.2. A deformation of X along the infinitesimal extension S → S′ is an affinespectral scheme X′ over S′ together with an isomorphism X→ X′×S′ S.

In other words, a deformation of X is a cartesian square:

X X′

S S′

p

2.10.3. From [Lur16a, Prop. 7.4.2.5], we have:

Lemma 2.10.4. The datum of a deformation of X along S → S′ is equivalent to the datum ofa null-homotopy of the composite

T∗X/S[−1]→ p∗(T∗S)→ p∗(F).

Given such a null-homotopy, one obtains a derivation d′ : T∗X → p∗(F); the deformation X′ isconstructed as the infinitesimal extension of X along d′.

2.10.5. For example, if p is smooth, then any morphism T∗X/S[−1] → p∗(F) must be null-

homotopic; hence X admits a deformation along any infinitesimal extension S → S′. If p isfurther etale, then this deformation is unique.

2.11. Push-outs of closed immersions.


2.11.1. Let i1 : Y → X1 and i2 : Y → X2 be closed immersions of spectral schemes.

The following is a straightforward variation on [Lur16b, Thm. 16.1.0.1] (cf. [Lur16b, Rem.16.1.0.2]):

Lemma 2.11.2.

(i) There a cocartesian square of spectral schemes

Y X1

X2 X,

i1

i2 k2

k1

where k1 and k2 are closed immersions.

(ii) If i1 (resp. i2) is a nil-immersion, then k1 (resp. k2) is a nil-immersion.

2.12. Lifting smooth morphisms along closed immersions.

2.12.1. The following is a spectral version of [Gro67, Prop. 18.1.1]:

Proposition 2.12.2. Let i : Z → S be a closed immersion of spectral schemes. For any smooth(resp. etale) morphism p : X → Z, there exists, Zariski-locally on X, a smooth (resp. etale)morphism q : Y → S, and a cartesian square

X Y

Z S.

p q

Proof. First we consider the etale case. The question being Zariski-local, we may assume that S,Z and X are affine. Consider the Postnikov towers (Proposition 2.7.5)

Scl = S60 → S61 → · · · → S6n → · · · → S

Zcl = Z60 → Z61 → · · · → Z6n → · · · → Z

for S and Z, respectively. Since p is flat, the Postnikov tower for X is identified with the basechange of the Postnikov tower of Z.

For a fixed integer n > 0, consider the following claim:

(∗) There exists, Zariski-locally on X6n, an etale morphism q6n : Y6n → S6n and a cartesiansquare

X6n Y6n

Z6n S6n.

p6n q6n

Note that it suffices to show that (∗) holds for each n > 0, since we can conclude by passingto filtered colimits. For n = 0, the claim is [Gro67, Prop. 18.1.1].

We proceed by induction; assume that the claim holds for a fixed n. We define Y6n+1 tobe the deformation of Y6n along the infinitesimal extension S6n → S6n+1, which exists byLemma 2.10.4. Note that X6n+1 is itself a deformation of X6n along the infinitesimal extensionZ6n → Z6n+1. That the resulting square is cartesian is a straightforward verification.

In the smooth case, the claim follows from the etale case and from Proposition 2.9.10.

2.13. Regular closed immersions.

16 ADEEL A. KHAN

2.13.1. Let i : Z → X be a closed immersion of spectral schemes. We define:

Definition 2.13.2. The closed immersion i is regular if the shifted cotangent sheaf TZ/X[−1]is a locally free OZ-module of finite rank.

We have (cf. [AG15, Prop. 2.1.10]):

Lemma 2.13.3. A closed immersion i : Z → X is regular if and only if, Zariski-locally on X,there exists a morphism f : X→ An and a cartesian square

Z X

Spec(S) An,

f

z

where z denotes the zero section.

2.13.4. Let i : Z → X be a regular closed immersion.

We define the conormal sheaf N∗Z/X as the shifted cotangent sheaf

N∗Z/X := T∗Z/X[−1].

By assumption, this is a locally free OZ-module of finite rank. The associated vector bundle(Definition 2.6.7), which we denote N∗Z/X, is called the conormal bundle of i.

2.13.5. The virtual codimension of i, defined Zariski-locally on Z, is the rank of the locally freeOZ-module N∗Z/X.

2.13.6. By Lemma 2.8.4 we have:

Lemma 2.13.7. Let i : Z → X be a closed immersion of smooth spectral S-schemes. Then i isregular.

2.13.8. The following is a slight variation on Lemma 2.13.3:

Lemma 2.13.9. Let p : X→ S be a smooth morphism of affine spectral schemes admitting asection s : S → X. Then there exists an S-morphism q : X → N∗S/X satisfying the following

conditions:

(i) There is a cartesian square

S X

S N∗S/X,

s

q

z

where z denotes the zero section.

(ii) The morphism q is etale on some Zariski neighbourhood X0 ⊂ X of s. That is, the morphisms factors through an open immersion j0 : X0 → X with q j0 etale.

Proof. Consider the closed immersion scl : Scl → Xcl of underlying classical schemes, which isdefined by a (discrete) quasi-coherent sheaf of ideals I. Since scl admits a retraction pcl, whichis smooth by Lemma 2.9.11, it is a regular closed immersion of classical schemes. In particularits conormal sheaf I/I2 is a (discrete) projective finitely generated OScl

-module.

Therefore the epimorphism (pcl)∗(I)→ I/I2 admits a section, which gives rise to a morphismI/I2 → (pcl)∗(OXcl

). A lift N∗s → p∗(OX) of this morphism corresponds to a morphism ofOS-algebras

ϕ : SymOS(N∗s)→ p∗(OX)


such that the commutative square of OS-algebras

SymOS(N∗s) OS

p∗(OX) OS

ζ

ϕ

σ

is cocartesian. Here ζ corresponds to the zero section z and σ to s. We let q : X→ N∗s be themorphism of spectral S-schemes corresponding to ϕ.

For (ii), let j0 be the etale locus of q. To show that s factors through j0, it is sufficient tonote that s∗(T∗X/N∗s

) ≈ T∗S/S ≈ 0 by (i).

3. Abstract Morel–Voevodsky homotopy theory

In this section we collect some generalities about Morel–Voevodsky homotopy theory in anabstract setting. We claim no originality for this material.

Given any essentially small ∞-category C, equipped with some additional structures, weconstruct unstable and stable homotopy categories associated to C.

Aside from classical motivic homotopy theory and its generalizations to derived and spectralalgebraic geometry, the setup also applies to interesting variants like equivariant motivichomotopy theory as developed in [Hoy17], and even noncommutative motivic homotopy theoryas developed in [Rob15].

3.1. Presheaves. In this short paragraph we fix our notations regarding ∞-categories ofpresheaves.

3.1.1. Let C be an essentially small ∞-category. We write PSh(C) for the ∞-category ofpresheaves of spaces on C, i.e. functors (C)op → Spc.

We will write hC : C → PSh(C) for the fully faithful functor defined by the Yonedaembedding.

3.1.2. Recall that PSh(C) is the free presentable ∞-category generated by the ∞-category C.

More precisely, for any presentable ∞-category D, let Funct!(PSh(C),D) denote the ∞-category of colimit-preserving functors PSh(C) → D. Then we have the following universalproperty ([Lur09, Thm. 5.1.5.6]):

Theorem 3.1.3. For any presentable ∞-category D, the canonical functor

(3.1) Funct!(PSh(C),D)→ Funct(C,D),

given by restriction along the Yoneda embedding hC : C → PSh(C), is an equivalence of∞-categories.

In particular, any functor u : C → D admits a unique extension to a colimit-preservingfunctor u! : PSh(C)→ D, called the left Kan extension of u.

3.1.4. Recall that the cartesian product defines a canonical closed symmetric monoidal structureon the category PSh(C).

3.2. Left localizations. We briefly recall the theory of left localizations of presentable ∞-categories, following [Lur09].

18 ADEEL A. KHAN

3.2.1. Let W be a set of morphisms in a presentable ∞-category A. We define:

Definition 3.2.2.

(i) An object c ∈ A is W-local if, for every morphism f : x→ y in W, the induced morphism ofspaces

MapsA(y, c)→ MapsA(x, c)

is invertible.

(ii) A morphism f : x→ y is a W-local equivalence if for every W-local object c, the inducedmorphism of spaces

MapsA(y, c)→ MapsA(x, c)

is invertible.

3.2.3. We use the following theorem repeatedly (see [Lur09, Prop. 5.5.4.15]):

Theorem 3.2.4. Let A be a presentable ∞-category. For any essentially small8 set W ofmorphisms in A, the inclusion of the full subcategory LWA ⊂ A of W-local objects admits a leftadjoint LW : A→ LWA, which exhibits LWA as an accessible left localization of A. Further, amorphism f in A induces an isomorphism LW(f) if and only if f is a W-local equivalence, orequivalently, if and only if f belongs to the strongly saturated closure of the set W.

Here, a left localization is by definition a functor that admits a fully faithful right adjoint.In fact, all accessible left localizations of a presentable ∞-category A arise in the above way.(Accessibility is a set-theoretic condition which we will not bother to explain here.)

3.2.5. Let A be a presentable ∞-category and W an essentially small set of morphisms.

Given a presentable ∞-category B, let Funct!,W(A,B) denote the full subcategory ofFunct!(A,B) spanned by functors that send morphisms in W to isomorphisms in B.

We have the following universal property of LWA (see [Lur09, Prop. 5.5.4.20]):

Theorem 3.2.6. For any presentable ∞-category B, the canonical morphism

Funct!(LWA,B)∼−→ Funct!,W(A,B)

given by restriction along the functor LW : A→ LWA, is an equivalence.

3.3. Pointed objects. We recall some standard facts about pointed objects in presentable∞-categories.

3.3.1. Let A be an presentable ∞-category. We write A• for the presentable ∞-category ofpointed objects in A. Its objects are pairs (a, x), where a is an object of A and x : pt→ a is amorphism from the terminal object.

By [Lur16a, Ex. 4.8.1.20, Prop. 4.8.2.11], A• has a canonical structure of Spc•-modulecategory, and is canonically equivalent to the base change A⊗Spc Spc•.

3.3.2. Consider the forgetful functor sending a pointed object (a, x) to its underlying objecta ∈ A. This admits a left adjoint, which freely adjoins a point to a; that is, it is given on objectsby the assignment

a 7→ a+ := (a t pt, x)

where x is the canonical morphism pt→ a t pt.

8A set of morphisms W is essentially small if there is a small subset W0 ⊂W such that every morphism inW is isomorphic to a morphism of W0 in the category of arrows of A.


3.3.3. Note that A• is equivalent to the category of modules over the monad with underlyingendofunctor a 7→ a t pt. Since the latter commutes with contractible colimits, this implies:

Lemma 3.3.4. The forgetful functor (a, x) 7→ a is conservative, and preserves and reflectscontractible colimits.

This monadic description also implies that every pointed object can be written as the geometricrealization of a simplicial diagram with each term in the essential image of a 7→ a+:

Lemma 3.3.5. The ∞-category A• is generated under sifted colimits by objects of the forma+, where a is an object of A.

3.3.6. We have the following universal property of A•:

Lemma 3.3.7. For any pointed presentable ∞-category D, the canonical functor

(3.2) Funct!(A•,D)→ Funct!(A,D),

given by restriction along the functor a 7→ a+, is an equivalence.

3.3.8. Suppose that A is the ∞-category of presheaves on some essentially small ∞-categoryC.

In this case, the ∞-category A• = PSh(C)• can be described as the free pointed presentable∞-category generated by the∞-category C. Indeed, combining Lemma 3.3.7 with Theorem 3.1.3we obtain a canonical equivalence

(3.3) Funct!(PSh(C)•,D)∼−→ Funct(C,D).

for any pointed presentable ∞-category D.

3.3.9. Suppose that A admits a closed symmetric monoidal structure.

Then the category A• inherits a canonical closed symmetric monoidal structure, given by the“smash product” ∧. The monoidal unit is the object pt+.

This monoidal structure is uniquely characterized by the fact that the functor a 7→ a+ issymmetric monoidal. Further, we have the following universal property (see [Rob15, Cor. 2.32]):

Lemma 3.3.10. For any pointed symmetric monoidal presentable ∞-category D, the canonicalfunctor

(3.4) Funct!,⊗(A•,D)→ Funct!,⊗(A,D),

given by restriction along the symmetric monoidal functor a 7→ a+, is an equivalence of ∞-categories.

3.3.11. Let t ∈ A• be a pointed object in A.

The t-suspension endofunctor Σt on A• is defined by the assignment

(a, x) 7→ (a, x) ∧ t.

Dually, the t-loop space endofunctor Ωt is given by

(a, x) 7→ Hom(t, (a, x)).

These endofunctors form an adjunction (Σt,Ωt).

3.4. Spectrum objects. We recall some standard facts about spectrum objects in pointedpresentable ∞-categories, following [Hoy17, §6.1].

20 ADEEL A. KHAN

3.4.1. Let A be a presentable∞-category. We fix an (essentially small) set T of pointed objectsof A.

We write SptT(A•) for the presentable ∞-category of T-spectrum objects in A•. We nowrecall its construction.

Suppose that the set T contains a finite number of elements t1, . . . , tn. Then write ΣT :=Σt1⊗···⊗tn , and define SptT(A•) as the colimit of the filtered diagram

(3.5) A•ΣT−−→ A•

ΣT−−→ · · ·

in the ∞-category of presentable ∞-categories. Equivalently, this is the limit of the cofiltereddiagram

(3.6) · · · ΩT−−→ A•ΩT−−→ A•

in the ∞-category of ∞-categories, where we have written ΩT := Ωt1⊗···⊗tn .

In the infinite case, we define SptT(A•) as the filtered colimit

SptT(A•) = lim−→T0⊂T

SptT0(A•),

indexed over finite subsets T0 ⊂ T.

Remark 3.4.2. Note that, in the case where T is finite, a T-spectrum is the data of a sequence(an)n>0 of pointed presheaves and structural isomorphisms

αn : an∼−→ Ωt(an+1)

for each integer n > 0.

In the infinite case, we have, roughly speaking, deloopings with respect to finite tensorproducts of elements of T, and a homotopy coherent system of compatibilities.

3.4.3. By construction, the adjunctions (Σt,Ωt) (3.3.11) at the level of pointed spaces give riseto equivalences

ΣT0 : SptT(A•) SptT(A•) : ΩT0 ,

for each finite subset T0 ⊂ T.

3.4.4. By construction, we have a canonical adjunction

Σ∞T : A• SptT(A•) : Ω∞T .

3.4.5. By [Lur09, Lem. 6.3.3.6], and the construction of SptT(A•), we have9:

Lemma 3.4.6.

(i) If T is finite, then the category SptT(A•) is generated under filtered colimits by objects of theform ΩnTΣ∞T (a), for a ∈ A• a pointed object and n > 0.

(ii) If T is infinite, then the category SptT(A•) is generated under filtered colimits by objectsof the form ΩTn · · ·ΩT0

Σ∞T (a), for a ∈ A• a pointed object, n > 0, and T0, . . . ,Tn ⊂ T finitesubsets.

9For the reader’s convenience, we state the finite case separately.


3.4.7. We have10 the following universal property of the presentable ∞-category SptT(A•):

Lemma 3.4.8. Let D be a presentable ∞-category.

(i) If T is finite, then the canonical functor

Funct!(SptT(A•),D)→ lim←−Funct!(A•,D)

is an equivalence.

(ii) If T is infinite, then the canonical functor

Funct!(SptT(A•),D)→ lim←−T0⊂T

lim←−Funct!(A•,D)

is an equivalence.

In the first statement, the limit is of the diagram

· · · (ΣT)∗−−−−→ Funct!(A•,D)(ΣT)∗−−−−→ Funct!(A•,D).

In the second, the first limit is taken over finite subsets T0 ⊂ T, and the second limit is of thediagram

· · ·(ΣT0

)∗

−−−−→ Funct!(A•,D)(ΣT0

)∗

−−−−→ Funct!(A•,D).

Proof. First consider the finite case. It suffices to show that, for every presentable ∞-categoryE, the map of spaces obtained by applying the functor Maps(E,−) is invertible. Recall thatFunct! computes the internal Hom in the closed symmetric monoidal ∞-category of presentable∞-categories, i.e. we have Maps(E,Funct!(E

′,E′′)) ≈ Maps(E⊗E′,E′′) for any two presentable∞-categories E′ and E′′. Also, recall the tensor product of presentable ∞-categories commuteswith colimits in each argument. It follows that, for each E, the map in question is the canonicalisomorphism

Maps(SptT(E⊗A•),D)∼−→ lim←−Maps(E⊗A•,D).

In the infinite case, we use an argument similar to the one above to reduce to the finitecase.

3.4.9. Suppose that each pointed object t ∈ T is k-symmetric, i.e. the cyclic permutation oft⊗k is homotopic to the identity morphism, for some k > 2.

In this case the main result of [Rob15] endows the presentable ∞-category SptT(A•) with acanonical closed symmetric monoidal structure.

This monoidal structure is uniquely characterized by the fact that the functor a 7→ Σ∞T (a)lifts to a symmetric monoidal functor that sends each object t ∈ T to a monoidally invertibleobject of SptT(A•).

The monoidal unit 1SptT(A•) is the T-spectrum Σ∞T (pt+).

Further, we have the following universal property:

Lemma 3.4.10. If each object t ∈ T is k-symmetric for some k > 2, then the canonical functor

Funct!,⊗(SptT(A•),D)→ Funct!,⊗,T(A•,D),

given by restriction along the functor a 7→ Σ∞T (a), is an equivalence.

Here Funct!,⊗,T(A•,D) denotes the full sub-∞-category of Funct!,⊗(A•,D) spanned bysymmetric monoidal colimit-preserving functors that send each object t ∈ T to a monoidallyinvertible object of D.

10For the reader’s convenience, we state the finite case separately.

22 ADEEL A. KHAN

3.5. Homotopy invariance.

3.5.1. Let A be an (essentially small) set of morphisms in C, which is stable under base change.That is, for each morphism a : c′ → c in A, and any morphism f : d→ c in C, the base changec′×c d→ d exists and belongs to A.

We define:

Definition 3.5.2. A presheaf F is A-invariant if for every morphism a : c′ → c in A, theinduced morphism of spaces

F(a) : F(c)→ F(c′)

is invertible.

Note that we have:

Lemma 3.5.3. The condition of A-invariance is stable by colimits.

Proof. Let F be the colimit of a diagram (Fα)α, where each Fα isA-invariant. For any morphisma ∈ A, the morphism F(a) is the colimit of the morphisms Fα(a), since colimits of presheavesare computed section-wise.

3.5.4. Let PShA(C) denote the full sub-∞-category of PSh(C) spanned by presheaves thatare A-invariant. This is the left localization at the small set A, and as such there exists alocalization functor F 7→ LA(F), left adjoint to the inclusion.

We say that a morphism of presheaves is an A-local equivalence if it becomes invertible afterapplying the functor LA. The set of A-local equivalences is equivalently the strongly saturatedclosure of the set A.

3.5.5. According to [Hoy17, Prop. 3.3], the basic properties of PShA(C) can be summarized asfollows:

Lemma 3.5.6.

(i) For each presheaf F on C, there is a canonical isomorphism

(3.7) LA(F)(c) ≈ lim−→(d→c)∈(Ac)op

F(d)

for each object c ∈ C, where Ac denotes the full sub-∞-category of C/c spanned by compositesof morphisms in the set A. Further, the category (Ac)op is sifted.

(ii) The functor F 7→ LA(F) commutes with finite products.

(iii) The category PShA(C) has universality of colimits.

3.6. Excision structures. The notion of excision structure is a slight variation on cd-structuresstudied by Voevodsky [Voe10].

3.6.1. A pre-excision structure on C is an essentially small set E of cartesian squares in C.

We may consider the following axioms:

(EXC0) There exists an initial object ∅C of C such that every morphism c→ ∅C is invertible.

(EXC1) The set E is stable under base change. That is, for any square Q in E, the basechange along any morphism in C exists and belongs to E.

Definition 3.6.2. We say that E is an excision structure if it satisfies the axioms (EXC0) and(EXC1).


3.6.3. Let E be a pre-excision structure on C. We define:

Definition 3.6.4. A presheaf F on C is E-excisive if it satisfies the following conditions:

(i) The presheaf F is reduced, i.e. the space F(∅C) is contractible.

(ii) For any cartesian square Q ∈ E of the form

(3.8)

d′ d

c′ c,

g

f

the induced commutative square of spaces

F(c) F(c′)

F(d) F(d′)

is cartesian.

3.6.5. Note that we have:

Lemma 3.6.6. The condition of E-excision is stable by filtered colimits.

Proof. Condition (i) follows from the fact that filtered colimits of contractible spaces arecontractible. Condition (ii) follows from the fact that finite limits commute with filtered colimitsin PSh(C).

3.6.7. Let PShE(C) denote the full sub-∞-category of PSh(C) spanned by E-excisive presheaves.

Note that this is a left localization of PSh(C) at the (essentially small) set containing thecanonical morphism

e : ∅PSh(C) → hC(∅C)

and the morphisms

(3.9) kQ : KQ → hC(c)

for all squares Q ∈ E of the form (3.8). Here KQ denotes the presheaf

KQ = hC(c′) thC(d′)

hC(d).

In particular, it follows that there exists a localization functor F → LE(F), left adjoint to theinclusion.

3.6.8. An E-local equivalence is a morphism of presheaves which becomes invertible afterapplying the localization functor LE. The set of E-local equivalences is equivalently the stronglysaturated closure of the set of morphisms containing e and kQ for each Q ∈ E.

3.7. Topological excision structures.

24 ADEEL A. KHAN

3.7.1. Let E be an excision structure on C. We consider the following axioms on E:

(EXC2) For every square of the form (3.8) in E, the lower horizontal morphism f is amonomorphism.

(EXC3) For every square in E of the form (3.8), the induced cartesian square

(3.10)

d′ d

d′×c′ d′ d×c d,

where the vertical arrows are the respective diagonal morphisms, belongs to E.

We define:

Definition 3.7.2. An excision structure E is topological if the axioms (EXC2) and (EXC3)are satisfied.

This terminology will be explained by Theorem 3.7.9.

3.7.3. Let E be an excision structure on C.

Consider the Grothendieck pretopology on C consisting of the following covering families:

(1) The empty family covering ∅C.

(2) For every square Q ∈ E of the form

d′ d

c′ c,

g

f

the family f, g covering c.

We let τE denote the Grothendieck topology generated by this pretopology.

3.7.4. Note that the axioms (EXC0) and (EXC1) imply that families of the form (1) and (2)are stable under pullback.

It follows from [Hoy15, Cor. C.2] that the condition of descent (Cech descent) with respectto the topology τE can be described as follows.

Lemma 3.7.5. A presheaf F on C is a τE-sheaf if it satisfies the following conditions:

(i) The space F(∅C) is contractible.

(ii) For every square Q ∈ E of the form (3.8), the canonical morphism of spaces

F(c)→ lim←−[n]∈∆

MapsPSh(C)(C(c/c)n,F)

is invertible. Here the simplicial object C(c/c)• is the Cech nerve of the morphism c = h(c′) th(d)→ h(c).

3.7.6. Let PShτE(C) denote the full sub-∞-category of PSh(C) spanned by τE-sheaves, i.e.presheaves satisfying τE-descent.

Note that this is the left localization at the essentially small set containing the canonicalmorphism

e : ∅PSh(C) → hC(∅C)


and the morphisms

(3.11) cQ : CQ → hC(c),

for all squares Q ∈ E of the form (3.8), where CQ denotes the presheaf

CQ = lim−→[n]∈∆op

C(c/c)n,

where c = h(c′) t h(d).

In particular, there is a localization functor F 7→ Lτ (F), left adjoint to the inclusion. By∞-topos theory, we have:

Proposition 3.7.7.

(i) The localization functor LτE is left exact, i.e. it commutes with finite limits.

(ii) The ∞-category ShτE(C) has universality of colimits.

A τE-local equivalence is a morphism of presheaves which becomes invertible after applying thelocalization functor LτE . The set of τE-local equivalences is equivalently the strongly saturatedclosure of the set of morphisms containing e and cQ for each Q ∈ E.

3.7.8. The following theorem of Voevodsky says that the localization defined by any topologicalexcision structure coincides with the localization defined by the associated Grothendieck topology.

Theorem 3.7.9 (Voevodsky). If E is a topological excision structure, then for any presheaf Fon C, the condition of E-excision is equivalent to τE-descent.

This was proved in [Voe10, Thm. 5.10] (cf. [AHW15, Thm. 3.2.5]) in the case where the∞-category C is an ordinary category. The reader will note that the proof generalizes mutatismutandis to our setting.

In particular we obtain:

Corollary 3.7.10. If E is a topological excision structure, then we have:

(i) The localization functor LE is left-exact, i.e. commutes with finite limits.

(ii) The ∞-category PShE(C) has universality of colimits.

3.8. Unstable homotopy theory.

3.8.1. Let C be an essentially small ∞-category, admitting an initial object ∅C.

We fix an excision structure E on C, and an (essentially small) set A of morphisms in C,which is stable under base change.

The unstable homotopy theory associated to the pair (E,A) is the ∞-category HE,A(C)defined as the full sub-∞-category of PSh(C) spanned by presheaves satisfying E-excision andA-invariance.

3.8.2. This is an accessible left localization of PSh(C), so HE,A(C) is a presentable∞-category,and the inclusion admits a left adjoint F 7→ LE,A(F).

A (E,A)-local equivalence is a morphism of presheaves that becomes invertible after applyingLE,A.

3.8.3. Let F be an E-excisive A-invariant presheaf. For any object c ∈ C, we have canonicalbifunctorial isomorphisms of spaces

MapsHE,A(C)(LE,A(hC(c)),F) ≈ F(c).

26 ADEEL A. KHAN

3.8.4. Though the localization functors LE and LA do not commute, the functor LE,A can bedescribed by the following transfinite composition:

Lemma 3.8.5. For every presheaf F on C, there is a canonical isomorphism

(3.12) LE,A(F) ≈ lim−→n>0

(LA LE)n(F).

Proof. This follows from the fact that the properties of E-excision and A-invariance are stableunder filtered colimits (Lemmas 3.6.6 and 3.5.3).

3.8.6. We have:

Corollary 3.8.7. If the excision structure E is topological, then the following hold.

(i) The localization functor LE,A commutes with finite products.

(ii) The ∞-category HE,A(C) has universality of colimits.

Proof. The first claim follows from the formula given in Lemma 3.8.5. Indeed, the functors LEand LA both commute with finite products (Lemmas 3.7.10 and 3.5.6), and filtered colimitscommute with finite products of presheaves.

The second claim follows directly from Corollary 3.7.10 and Lemma 3.5.6.

3.8.8. We now formulate a universal property for HE,A(C).

For a presentable∞-category D, let FunctE,A(C,D) denote the full subcategory of Funct(C,D)spanned by functors u : C→ D that satisfy E-excision and A-invariance, i.e. which send

(1) the initial object ∅C to an initial object of D,

(2) any square Q ∈ E to a cocartesian square in D,

(3) any morphism a ∈ A to an invertible morphism in D.

Then we have:

Lemma 3.8.9. For any presentable ∞-category D, the canonical functor

(3.13) Funct!(HE,A(C),D)→ FunctE,A(C,D),

given by restriction along the functor C→ HE,A(C), is an equivalence of ∞-categories.

Proof. This follows immediately from Theorem 3.1.3 and the universal property of left localiza-tions (Theorem 3.2.6).

3.8.10. By point (i) of Corollary 3.8.7, we have:

Corollary 3.8.11. Suppose that the excision structure E is topological. Then the cartesianmonoidal structure on the presentable ∞-category PSh(C) restricts to a cartesian monoidalstructure on the presentable ∞-category HE,A(C).

3.9. Pointed homotopy theory.

3.9.1. Let PSh(C)• denote the presentable ∞-category of pointed objects in the presentable∞-category PSh(C).

For any pointed presheaf (F, x) on C, we define:

Definition 3.9.2. The pointed presheaf (F, x) is E-excisive or A-invariant if its underlyingpresheaf F has the respective property.


Let HE,A(C)• denote the full sub-∞-category of PSh(C)• spanned by E-excisive A-invariantpointed presheaves. Note that this is equivalent to the presentable∞-category of pointed objectsin the presentable ∞-category HE,A(C).

We call this the pointed homotopy theory associated to the pair (E,A).

3.9.3. Note that HE,A(C)• is an accessible left localization of PSh(C) at the essentially smallset of morphisms of the form

(3.14) hC(a)+ : hC(c)+ → hC(c′)+,

for each morphism a : c→ c′ in A,

(3.15) (kQ)+ : (KQ)+ → hC(c)+,

for each square Q ∈ E of the form (3.8), and

e+ : pt→ hC(∅C)+.

3.9.4. In particular, we have localization functors LE, LA, and

(3.16) LE,A : PSh(C)• → HE,A(C)•

at the level of pointed objects.

By Lemma 3.3.10 it follows that these functors are symmetric monoidal, and are uniquelycharacterized by commutativity with the functor F 7→ F+.

3.9.5. Combining Lemma 3.8.9 with Lemma 3.3.7, we obtain the following universal propertyfor the pointed presentable ∞-category HE,A(C)•:


(3.17) Funct!(HE,A(C)•,D)∼−→ FunctE,A(C,D).

is an equivalence of ∞-categories.

3.9.7. By Lemma 3.3.10, we have a canonical closed symmetric monoidal structure on HE,A(C)•,characterized by the following universal property:

Lemma 3.9.8. For any pointed symmetric monoidal presentable ∞-category D, the canonicalfunctor

(3.18) Funct!,⊗(HE,A(C)•,D)→ Funct!,⊗(HE,A(C),D),

given by restriction along the functor F 7→ F+, is an equivalence of ∞-categories.

Here Funct!,⊗ denotes the ∞-category of colimit-preserving symmetric monoidal functors.

3.10. Stable homotopy theory.

3.10.1. We fix an (essentially small) set T of pointed presheaves on C.

We will always assume that each T ∈ T is E-excisive and A-invariant, by replacing it withits (E,A)-localization if necessary.

We will consider the following axioms on T:

(STAB1) At least one of the pointed presheaves T ∈ T can be written as an S1-suspensionΣS1(T′) of some pointed presheaf T′.

(STAB2) Each pointed presheaf T ∈ T is k-symmetric, i.e. the cyclic permutation of T⊗k ishomotopic to the identity morphism, for some k > 2.

28 ADEEL A. KHAN

3.10.2. Let SptT(PSh(C)•) denote the presentable ∞-category of T-spectrum objects (Para-graph 3.4) in PSh(C)•.

3.10.3. Since a pointed ∞-category is stable if and only if the S1-suspension functor ΣS1 is anequivalence, we have:

Lemma 3.10.4. If the axiom (STAB1) holds, then the presentable ∞-category SptT(PSh(C)•)is stable.

3.10.5. We define:

Definition 3.10.6. We say that a T-spectrum F satisfies E-excision or A-invariance if foreach n > 0, its nth component Ω∞−nT (F) satisfies the respective property (as a pointed presheaf).

Let SHE,A,T(C) denote the full sub-∞-category of SptT(PSh(C)•) spanned by E-excisiveA-invariant T-spectra. This is equivalent to the presentable ∞-category of T-spectrum objectsin the presentable ∞-category HE,A(C)•.

We call the presentable ∞-category SHE,A,T(C) the stable homotopy theory of C, withrespect to (E,A,T).

3.10.7. As in Lemma 3.10.4, we have:

Lemma 3.10.8. If the axiom (STAB1) holds, then the presentable ∞-category SHE,A,T(C) isstable.

3.10.9. Note that SHE,A,T(C) is the left localization at the essentially small set of morphismsof the form

(3.19) Σ∞T hC(a)+ : Σ∞T hC(c)+ → Σ∞T hC(c′)+,

for each morphism a : c→ c′ in A,

(3.20) Σ∞T (kQ)+ : Σ∞T (KQ)+ → Σ∞T hC(c)+,

for each square Q ∈ E of the form (3.8), and

Σ∞T (e+) : 0→ Σ∞T hC(∅C)+,

where 0 denotes the zero object of SptT(PSh(C)•).

In particular, we have localization functors LE, LA, and

(3.21) LE,A : SptT(PSh(C)•)→ SHE,A,T(C),

at the level of T-spectra.

3.10.10. Combining Lemma 3.9.6 with Lemma 3.4.8, we obtain the following universal propertyfor the presentable ∞-category SHE,A,T(C):


Funct!(SHE,A,T(C),D)→ lim←−T0⊂T

lim←−FunctE,A(C,D)

is an equivalence.


3.10.12. Suppose that the axiom (STAB2) holds. Then by Lemma 3.4.10, we have a canonicalclosed symmetric monoidal structure on SHE,A,T(C).

As a symmetric monoidal presentable ∞-category, SHE,A,T(C) is characterized by thefollowing universal property:

Lemma 3.10.13. Suppose the axiom (STAB2) holds. For any presentable ∞-category D, thecanonical functor

Funct!,⊗(SHE,A,T(C),D)→ Funct!,⊗(HE,A(C),D),

given by restriction along the functor F 7→ Σ∞T (F+), is an equivalence.

This follows by combining Lemma 3.4.10 with Lemma 3.9.8.

3.10.14. Using the universal property (Lemma 3.4.10), we see that the localization functors LE,LA, and LE,A are symmetric monoidal, and can be characterized uniquely by commutativitywith the functor Σ∞T .

4. Brave new motivic homotopy theory

In this section, we construct the brave new motivic homotopy category over any (quasi-compactquasi-separated) spectral base scheme S.

We write SmE∞/S for the (essentially small) category of smooth spectral schemes (of finite

presentation) over S.

A SmE∞-fibred space over S is a presheaf of spaces on SmE∞/S . When there is no risk of

confusion we will simply say fibred space or even space over S. We write SpcE∞(S) for the

category of SmE∞ -fibred spaces over S, and hS for the Yoneda embedding SmE∞/S → SpcE∞(S).

4.1. Nisnevich excision.

4.1.1. Let S be an spectral scheme. A Nisnevich square over S is a cartesian square of spectralschemes over S

(4.1)

U×X V V

U X

k

q p

j

such that j is an open immersion, p is etale, and there exists a closed immersion Z → Xcomplementary to j such that the induced morphism p−1(Z)→ Z is invertible.

Remark 4.1.2. Note that a commutative square of classical schemes is Nisnevich in the usual senseif and only if it induces a Nisnevich square of spectral schemes. This follows from Lemma 2.9.4.

4.1.3. We let ENisS denote the set of Nisnevich squares in the category SmE∞

/S . It is clear that

this defines an excision structure in the sense of Definition 3.6.2.

The following technical lemma will be useful:

Lemma 4.1.4. The excision structure ENisS is topological in the sense of Definition 3.7.2.

Proof. The axiom (EXC2) is clear, since every open immersion of spectral schemes is a monomor-phism.

30 ADEEL A. KHAN

For (EXC3), let Q be a Nisnevich square of the form (4.1) and consider the induced cartesiansquare

W V

W×U W V×X V.

k

∆W/U ∆V/X

(k,k)

It is clear that the lower horizontal morphism is an open immersion. The morphism ∆V/X isetale (in fact an open immersion) because p is etale, and it is clear that it induces an isomorphismon the complementary reduced closed subscheme.

4.1.5. We say that a SmE∞ -fibred space F is Nisnevich-local, or satisfies Nisnevich excision, ifthe following conditions hold:

(1) The presheaf F is reduced, i.e. the space F(∅) is contractible, where ∅ is the emptyspectral scheme.

(2) For every Nisnevich square of the form (4.1), the induced commutative square of spaces

Γ(X,F) Γ(U,F)

Γ(V,F) Γ(U×X V,F)

is cartesian.

We let SpcE∞Nis (S) denote the full subcategory of SpcE∞(S) spanned by Nisnevich-local spaces.

4.1.6. We have:

Proposition 4.1.7.

(i) The category SpcE∞Nis (S) is an accessible left localization of the category SpcE∞(S), i.e. there is

a localization functor LNis : SpcE∞(S)→ SpcE∞Nis (S), left adjoint to the inclusion. In particular,

SpcE∞Nis (S) is a presentable ∞-category.

(ii) The localization functor F 7→ LNis(F) is left-exact, i.e. commutes with finite limits. In

particular, SpcE∞Nis (S) is a topos, and has universality of colimits.

(iii) The full subcategory SpcE∞Nis (S) ⊂ SpcE∞(S) is stable under filtered colimits.

Proof. Claims (i) and (ii) follow from Corollary 3.7.10, in view of Lemma 4.1.4.

Claim (iii) follows from Lemma 3.6.6.

We will say that a morphism of SmE∞-fibred spaces is a Nisnevich-local equivalence if itbecomes invertible after applying the localization functor LNis.

Remark 4.1.8. Theorem 3.7.9 implies that the property of Nisnevich excision is equivalent toCech descent with respect to the Grothendieck topology generated by Nisnevich squares. Itfollows from [Lur16b, Thm. 3.7.5.1] that, over quasi-compact quasi-separated spectral schemes,this Grothendieck topology coincides with the Nisnevich topology as constructed by Lurie in[Lur16b, §3.7].

Since we do not assume that our spectral schemes are noetherian and finite-dimensional, thisdescent condition is in general much weaker than the condition of hyperdescent, i.e. descentwith respect to arbitrary hypercovers. We refer to [Lur09, §6.5.4] for an explanation of thisdistinction.


4.1.9. The following lemma follows from [Lur09, Prop. 5.5.8.10, (3)]:

Lemma 4.1.10. Let (Xα)α be a finite family of smooth spectral schemes over S. Then thecanonical morphism of presheaves

tα

hS(Xα)→ hS(tα

Xα)

is a Nisnevich-local equivalence.

By [Lur09, Lem. 5.5.8.14] it follows that the category of Nisnevich sheaves is generated undersifted colimits by the representables. In fact, we can say even more:

Lemma 4.1.11. The category of Nisnevich sheaves is generated under sifted colimits by therepresentable presheaves hS(X), where X = Spec(A) is an affine spectral scheme which is smoothover S.

Proof. Let hS(X) be a representable presheaf over S. Since hS(X) satisfies Nisnevich descent, wecan assume X is separated over S, by choosing an affine Zariski cover of X where the pairwiseintersections are separated. Then we repeat the same argument to assume X is affine, bychoosing an affine cover where the pairwise intersections are affine.

4.2. Homotopy invariance.

4.2.1. We say that a SmE∞-fibred space F over S satisfies A1-homotopy invariance, or isA1-local, if for every smooth spectral scheme X over S, the morphism of spaces

Γ(X,F)→ Γ(X×A1,F),

induced by the projection X×A1 → X, is invertible.

Here A1 denotes the spectral affine line over the sphere spectrum, i.e. A1 = Spec(St)(2.6.10).

4.2.2. Let SpcE∞A1 (S) denote the full subcategory of SpcE∞(S) spanned by A1-homotopy invariantspaces.

We have:

Lemma 4.2.3.

(i) The category SpcE∞A1 (S) is an accessible left localization of the category SpcE∞(S), i.e. there

is a localization functor LA1 : SpcE∞(S)→ SpcE∞A1 (S), left adjoint to the inclusion. In particular,

SpcE∞A1 (S) is a presentable ∞-category.

(ii) For every SmE∞-fibred space F, there is a canonical isomorphism

(4.2) Γ(X,LA1(F)) ≈ lim−→(Y→X)∈(AX)op

Γ(Y,F)

for each smooth spectral scheme X over S. Here (AX)op is a sifted small category, opposite to

the full subcategory of SmE∞/X spanned by compositions of A1-projections.

(iii) The localization functor F 7→ LA1(F) commutes with finite products.

(iv) The category SpcE∞A1 (S) has universality of colimits.

(v) The full subcategory SpcE∞A1 (S) ⊂ SpcE∞(S) is stable under colimits.

32 ADEEL A. KHAN

Proof. Let A denote the set of projections X×A1 → X for each smooth spectral scheme X overS; note that the condition of A1-homotopy invariance is nothing else than A-invariance in thesense of Definition 3.5.2.

Hence the claims follow from Lemma 3.5.3 and Lemma 3.5.6.

We will say that a morphism of SmE∞-fibred spaces is an A1-homotopy equivalence if itbecomes invertible after applying the A1-localization functor LA1 .

4.2.4. Let p : A1 → Spec(S) denote the projection to the terminal spectral scheme. Thisadmits two sections i0 and i1, the zero and unit sections, which are closed immersions of spectralschemes. They correspond to the morphisms St → S sending t to 0 and 1, respectively (whichare defined, as usual, up to a contractible space of choices).

Remark 4.2.5. Following [MV99, §2.3], it is possible to use the sections i0 and i1 to construct acosimplicial diagram of spectral schemes given degree-wise by

∆pS ≈ S×Ap ≈ S× Spec(St1, . . . , tp).

for each [p] ∈∆.

The localization functor LA1 can then be computed by the formula

(4.3) Γ(X,LA1(F)) ≈ lim−→[p]∈∆op

Γ(X×∆pS,F).

For our purposes, the formula (4.2) will suffice; both indexing categories (AX)op and ∆op aresifted, which is the only property we need.

4.2.6. Given two morphisms f, g : F ⇒ G of presheaves on SmE∞/S , an elementary A1-homotopy

from f to g is a morphism

hS(A1S)× F → G

whose restriction to F ≈ hS(S)× F along i0 (resp. i1) is isomorphic to f (resp. g).

We say that f and g are A1-homotopic if there exists a sequence of elementary A1-homotopiesconnecting them. In this case the induced morphisms LA1(F)⇒ LA1(G) coincide.

4.2.7. A morphism of presheaves ϕ : F → G is called a strict A1-homotopy equivalence if thereexists a morphism ψ : G→ F such that the composites ϕ ψ and ψ ϕ are A1-homotopic to theidentities.

Note that any strict A1-homotopy equivalence is an A1-homotopy equivalence.

4.3. Motivic spaces.

4.3.1. We define:

Definition 4.3.2. A motivic space over S is a SmE∞-fibred space F satisfying Nisnevich excisionand A1-homotopy invariance.

Let HE∞(S) denote the full subcategory of SpcE∞(S) spanned by motivic spaces.

4.3.3. For any smooth spectral scheme X over S, we let MS(X) := Lmot(hS(X)). We havecanonical bifunctorial isomorphisms

MapsHE∞ (S)(MS(X),F) ≈ Γ(X,F)

for every motivic space F over S.


4.3.4. We have:

Lemma 4.3.5.

(i) The category HE∞(S) is an accessible left localization of SpcE∞(S), i.e. there is a localization

functor Lmot : SpcE∞(S)→ HE∞(S), left adjoint to the inclusion. In particular, HE∞(S) is apresentable ∞-category.

(ii) The presentable ∞-category HE∞(S) admits a cartesian monoidal structure, and the localiza-tion functor F 7→ Lmot(F) lifts to a symmetric monoidal functor (i.e., it commutes with finiteproducts).

(iii) The localization functor F 7→ Lmot(F) can be described as the transfinite composite

(4.4) Lmot(F) ≈ lim−→n>0

(LA1 LNis)n(F).

(iv) The full subcategory HE∞(S) ⊂ SpcE∞(S) is stable under filtered colimits.

(v) The category HE∞(S) has the property of universality of colimits.

(vi) The category HE∞(S) is generated under sifted colimits by fibred spaces of the form MS(X),where X = Spec(A) is an affine spectral scheme which is smooth over S.

Proof. Note that the category HE∞(S) is nothing else than the category HENisS ,AS

(SmE∞/S ), the

unstable homotopy theory associated to the Nisnevich excision structure ENisS and the set of

A1-projections X×A1 → X (for X a smooth spectral S-scheme). Hence claims (i)–(v) followfrom the general results collected in Sect. 3.

Claim (vi) follows directly from Lemma 4.1.11.

We will say that a morphism of SmE∞ -fibred spaces is a motivic equivalence if it induces anisomorphism after applying the motivic localization functor Lmot.

4.4. Pointed motivic spaces.

4.4.1. Let SpcE∞(S)• denote the presentable ∞-category of pointed objects in SpcE∞(S), andHE∞(S)• the presentable ∞-category of pointed objects in HE∞(S).

We have:

Lemma 4.4.2.

(i) The category HE∞(S)• is an accessible left localization of SpcE∞(S)•, i.e. there is a localization

functor Lmot : SpcE∞(S)• → HE∞(S)•, left adjoint to the inclusion.

(ii) The presentable ∞-category HE∞(S)• admits a symmetric monoidal structure. The functorsF 7→ F+ and F 7→ Lmot(F) lift to symmetric monoidal functors.

(iii) The localization functor F 7→ Lmot(F) satisfies the formula

(4.5) Lmot(F+) ≈ Lmot(F)+

for every space F over S.

(iv) The category HE∞(S)• is generated under sifted colimits by objects of the form MS(X)+,where X = Spec(A) is an affine spectral scheme which is smooth over S.

Proof. The claims (i)–(iii) follow from the general statements of Paragraph 3.9.

Claim (iv) follows from point (vi) of Lemma 4.3.5 and Lemma 3.3.5.

34 ADEEL A. KHAN

We will write ∧S for the monoidal product on HE∞(S)•.

4.5. Motivic spectra.

4.5.1. Let SptE∞(S)S1 denote the stable presentable ∞-category SptS1(SpcE∞(S)•) of S1-

spectrum objects in the presentable ∞-category of pointed SmE∞-fibred spaces over S. (Herewe view S1 as a constant pointed presheaf.) This is equivalent to the presentable ∞-category of

presheaves of spectra on SmE∞/S .

The objects of SptE∞(S)S1 , which we will call (SmE∞-fibred) spectra over S, are sequences

(Fn)n>0 of pointed SmE∞-fibred spaces Fn, together with isomorphisms

Fn∼−→ ΩS1(Fn+1).

4.5.2. Let SHE∞(S)S1 denote the presentable ∞-category SptS1(HE∞(S)•). This is equivalent

to the presentable ∞-category of presheaves of spectra on SmE∞/S satisfying Nisnevich excision

and A1-homotopy invariance.

There is a canonical adjunction

(4.6) Σ∞S1 : HE∞(S)• SHE∞(S)S1 : Ω∞S1 .

We have:

Lemma 4.5.3.

(i) The category SHE∞(S)S1 is an accessible left localization of SptE∞(S)S1 , i.e. there is a

localization functor Lmot : SptE∞(S)S1 → SHE∞(S)S1 , left adjoint to the inclusion. In particularit is a presentable ∞-category.

(ii) The presentable ∞-category SHE∞(S)S1 is symmetric monoidal, and the functors Σ∞T andLmot lift to symmetric monoidal functors.

(iii) The presentable ∞-category SHE∞(S)S1 is stable, and the localization functor F 7→ Lmot(F)is exact.

(iv) The localization functor F 7→ Lmot(F) satisfies the formula

Lmot(Σ∞−kS1 (hS(X)+)) ≈ Σ∞−kS1 (MS(X)+)

for each smooth spectral scheme X over S.

(v) The localization functor F 7→ Lmot(F) can be computed as the composite

(4.7) Lmot ≈ LA1 LNis.

(vi) The presentable ∞-category SHE∞(S)S1 is generated under sifted colimits by objects of theform Σ∞−nS1 (MS(X)+), where X = Spec(A) is an affine spectral scheme which is smooth over S,and n > 0.

Proof. The claims follow from the general results of Paragraph 3.10 and point (iv) of Lemma 4.4.2.

5. Inverse and direct image functoriality

5.1. For motivic spaces.


5.1.1. Let f : T→ S be a morphism of spectral schemes. The direct image functor

(5.1) fSpc∗ : SpcE∞(T)→ SpcE∞(S)

is defined as restriction along the base change functor SmE∞/S → SmE∞

/T .

According to Theorem 3.1.3, its left adjoint f∗Spc, the inverse image functor, is uniquelycharacterized by commutativity with colimits and the formula

(5.2) f∗Spc(hS(X)) ≈ hT(X×S

T)

for smooth spectral schemes X over S.

5.1.2. Note that the base change functor SmE∞/S → SmE∞

/T preserves Nisnevich covering families

and A1-projections. It follows that the inverse image functor f∗Spc preserves Nisnevich-local

equivalences and A1-homotopy equivalences.

By adjunction, its right adjoint fSpc∗ preserves Nisnevich sheaves and A1-homotopy invariant

spaces and induces a functor fH∗ : HE∞(T)→ HE∞(S). This admits a left adjoint f∗H given by

the formulaf∗H(F) ≈ Lmot(f

∗Spc(F)).

According to Lemma 3.8.9, this is characterized by commutativity with colimits and the formula

(5.3) f∗H(MS(X)) ≈ MT(X×S

T).

5.1.3. Both the direct and inverse image functors are symmetric monoidal:

Lemma 5.1.4. The functor fSpc∗ (resp. fH

∗ ) admits a canonical symmetric monoidal structure.

Proof. Since the respective symmetric monoidal structures are cartesian, it suffices to show thatf∗ commutes with finite products. In fact, it commutes with arbitrary limits since it is a rightadjoint.

Lemma 5.1.5. The functor f∗Spc (resp. f∗H) admits a canonical symmetric monoidal structure.

Proof. By adjunction from Lemma 5.1.4, we obtain a canonical structure of colax symmetricmonoidal functor on f∗. That is, there are canonical morphisms

(5.4) f∗(F×SG)→ f∗(F)×

Tf∗(G)

for any two spaces F and G over S. It suffices to show that these morphisms are invertible.

Since f∗Spc commutes with colimits, and the cartesian product commutes with colimits in eachargument, one reduces to the case of representables, in which case the claim is clear. For f∗H,the claim follows from the first because motivic localization commutes with finite products.

5.2. For pointed motivic spaces.

5.2.1. Let f : T → S be a morphism of spectral schemes. Since fSpc∗ preserves the terminal

object, it induces a functor fSpc•∗ : SpcE∞(T)• → SpcE∞(S)• given on objects by the formula

fSpc•∗ (G, y) ≈ (fSpc

∗ (G), fSpc∗ (y)).

Its left adjoint f∗Spc•: SpcE∞(T)• → SpcE∞(S)• is uniquely characterized, according to

Lemma 3.3.5, by the fact that it commutes with sifted colimits and with the functor F 7→ F+:

(5.5) f∗Spc•(F+) ≈ f∗Spc(F)+

for any space F over S.

36 ADEEL A. KHAN

Explicitly, it is given on objects by the formula

f∗Spc•(F, x) ≈ (f∗Spc(F), f∗Spc(x))

for each pointed space (F, x) over S.

5.2.2. The direct image functor fSpc•∗ preserves the properties of Nisnevich descent and A1-

homotopy invariance, and induces a functor fH∗ . Its left adjoint f∗H• is given by composing f∗Spc•

with the motivic localization functor:

(5.6) f∗H• := Lmotf∗Spc•

.

It is uniquely characterized, according to Lemma 3.3.5, by commutativity with sifted colimitsand the formula

(5.7) f∗H•(F+) ≈ f∗H(F)+.

5.2.3.

Lemma 5.2.4. The inverse image functor f∗Spc•(resp. f∗H•) admits a canonical symmetric

monoidal structure.

Proof. For f∗Spc•, this follows directly from the universal property of Lemma 3.3.10 and the

formula (5.5). For f∗H• it follows from the universal property of Lemma 3.9.8 and the formula(5.7).

5.3. For motivic spectra.

5.3.1. Let f : T→ S be a morphism of spectral schemes.

By construction of SptE∞(S)S1 , there exists a unique functor f∗Spt : SptE∞(S)S1 → SptE∞(T)S1

which commutes with colimits and with the functor Σ∞S1 , i.e.:

(5.8) f∗SptΣ∞S1 ≈ Σ∞S1f∗Spc•

.

5.3.2. Let fSpt∗ be the right adjoint of f∗Spt. This can be described as the unique functor which

commutes with limits and with the functor Ω∞, i.e.:

(5.9) Ω∞fSpt∗ ≈ fSpc•

∗ Ω∞

It is given on objects by the assignment

E = (Fn)n 7→ f∗(E) = (fSpc•∗ (Fn))n.

5.3.3. The direct image functor fSpt∗ preserves motivic spectra and induces a functor fSH

∗ .

We let f∗SH be its left adjoint, the symmetric monoidal functor Lmotf∗Spt. This is the unique

functor which commutes with colimits and with the functor Σ∞S1 , i.e.:

(5.10) f∗SHΣ∞S1 ≈ Σ∞S1f∗H• .

5.3.4. Using the universal properties of Lemma 3.4.10 and Lemma 3.10.13, we get:

Lemma 5.3.5. The functor f∗Spt (resp. f∗SH) admits a canonical symmetric monoidal structure.

6. Functoriality along smooth morphisms

As per our conventions, smooth morphisms will be assumed to be of finite presentation.

6.1. The functor p].


6.1.1. Let p : X→ S be a smooth morphism of spectral schemes. In this case the base changefunctor admits a right adjoint, the forgetful functor SmE∞

/X → SmE∞/S :

(Y → X) 7→ (Y → Xp−→ S).

It follows that the functor p∗Spc coincides with restriction along the forgetful functor, andadmits a left adjoint

pSpc] : SpcE∞(T)→ SpcE∞(S).

This is uniquely characterized, according to Theorem 3.1.3, by commutativity with colimits andthe formula

(6.1) pSpc] (hX(Y)) ≈ hS(Y).

for smooth spectral schemes Y over X.

6.1.2. Since the forgetful functor SmE∞/X → SmE∞

/S preserves Nisnevich covering families and

A1-projections, it follows that pSpc] preserves Nisnevich-local equivalences and A1-homotopy

equivalences.

In particular its right adjoint p∗Spc preserves Nisnevich descent and A1-homotopy invariance,and induces a morphism p∗H on motivic spaces.

Its left adjoint pH] is given by applying pSpc

] and then the localization functor Lmot:

pH] (F) ≈ Lmot(p

Spc] (F)).

It is uniquely characterized, according to Lemma 3.8.9, by commutativity with colimits and theformula

(6.2) pH] (MX(Y)) ≈ MS(Y),

for smooth spectral schemes Y over X.

6.1.3. Let p : X→ S be a smooth morphism. By Lemma 5.1.5, the functors p∗Spc and p∗H admit

canonical symmetric monoidal structures, so that their respective left adjoints pSpc] and pH

]

admit colax symmetric monoidal structures.

If p is an open immersion, then these monoidal structures are strict:

Lemma 6.1.4. Let j : U → X be a quasi-compact open immersion. Then the canonical colax

symmetric monoidal structure on the functor jSpc] (resp. jH] ) is strict.

Proof. It suffices to show that the canonical morphisms

jSpc] (F×

UG)→ jSpc

] (F)×SjSpc] (G)

are invertible for all spaces F and G on U. Since jSpc] commutes with colimits, and the cartesian

product commutes with colimits in each argument, one reduces to the case of representables.

Then the claim follows from the fact that the fibred products X×U Y and X×S Y arecanonically identified (since j is a monomorphism, i.e. its diagonal morphism is invertible). Theclaim for jH] follows from the fact that motivic localization commutes with finite products.

38 ADEEL A. KHAN

6.1.5. Let (fα : Sα → S)α be a Nisnevich covering family. Given a morphism of motivic spacesover S, the following proposition says that it is invertible if and only if its inverse image on eachSα is invertible:

Proposition 6.1.6 (Nisnevich separation). Let S be a spectral scheme. For any Nisnevichcovering family (pα : Sα → S)α, the family of inverse image functors (pα)∗H : HE∞(S) →HE∞(Sα) is conservative.

This is in fact true at the level of Nisnevich sheaves, which is what we will prove.

Proof. Let ϕ : F1 → F2 be a morphism of Nisnevich sheaves on S, and suppose that the followingcondition holds:

(∗) For each α, the morphism (pα)∗Nis(F1)→ (pα)∗Nis(F2) is invertible.

The claim is that under this assumption, the morphism

Γ(X,F1)→ Γ(X,F2)

is invertible for every smooth spectral S-scheme X.

Since Fi satisfy Nisnevich descent, it suffices to show that the morphism

(6.3) Γ(Xα,F1)→ Γ(Xα,F2)

is invertible for each α, where Xα is the base change of X along pα.

Since hS(Xα) ≈ (pα)](pα)∗(hS(X)), we have by adjunction

Γ(Xα,Fi) ≈ Γ(X, (pα)](pα)∗Fi)

for each α and i.

Hence the claim follows from the assumption (∗).

6.1.7. As in Paragraph 5.2, the functor p] extends immediately to pointed spaces.

That is, we obtain functors

pSpc•] : SpcE∞(X)• → SpcE∞(S)•,

pH•] : HE∞(X)• → HE∞(S)•

left adjoint to p∗Spc•and p∗H• , respectively. These are uniquely characterized by commutativity

with colimits and the formulas

(6.4) pSpc•] (hX(Y)+) ≈ hS(Y)+, pH•

] (MX(Y)+) ≈ MS(Y)+.

6.1.8. Similarly, as in Paragraph 5.3, the functor p] extends immediately to spectra.

That is, we obtain functors

pSpt] : SptE∞(X)S1 → SptE∞(S)S1 ,

pSH] : SHE∞(X)S1 → SHE∞(S)S1

left adjoint to p∗Spt and p∗SH, respectively. These are uniquely characterized by commutativitywith colimits and the formulas

(6.5) pSpt] (Σ∞S1 hX(Y)+) ≈ Σ∞S1 hS(Y)+, pSH

] (Σ∞S1 MX(Y)+) ≈ Σ∞S1 MS(Y)+.

6.2. Smooth base change formulas.


6.2.1. Suppose we have a cartesian square

T′ S′

T S

f ′

p′ p

f

of spectral schemes.

At the level of (motivic) spaces, pointed spaces, and spectra, there are canonical 2-morphisms

(p′)](f′)∗ → f∗p],(6.6)

p∗f∗ → (f ′)∗(p′)∗,(6.7)

constructed in Paragraph A.4.

The following says that SpcE∞ and HE∞ satisfy the left base change property along smoothmorphisms (see loc. cit.):

Proposition 6.2.2. If p and p′ are smooth, then the 2-morphisms (6.6) and (6.7) are invertibleat the level of spaces and motivic spaces.

Proof. It suffices to consider (6.6); the morphism (6.7) is its right transpose.

At the level of fibred spaces, we note that the functors in question commute with colimits, sothat we may reduce to representable spaces, in which case the claim is obvious.

Similarly, for motivic spaces we may reduce to the case of motivic localizations of representablespaces.

6.2.3. Next we consider the case of pointed spaces. Then we have:

Proposition 6.2.4. If p and p′ are smooth, then the 2-morphisms (6.6) and (6.7) are invertibleat the level of pointed spaces and pointed motivic spaces.

Proof. By transposition it suffices to consider (6.6). Since the functors in question commutewith colimits and with the functor F 7→ F+, the claim follows from Lemma 3.3.5 and smoothbase change for unpointed spaces (Proposition 6.2.2).

6.2.5. At the level of spectra, we have:

Proposition 6.2.6. If p and p′ are smooth, then the 2-morphisms (6.6) and (6.7) are invertibleat the level of spectra and motivic spectra.

Proof. This follows from Lemma 3.4.6 and smooth base change for pointed spaces (Proposi-tion 6.2.4).

6.3. Smooth projection formulas. Let p : X→ S be a smooth morphism.

As recalled in Definition A.4.6, there are canonical morphisms

(6.8) pSpc] (F × p∗Spc(G))→ pSpc

] (F)× G

for any pair of SmE∞-fibred spaces F ∈ SpcE∞(X) and G ∈ SpcE∞(S); by adjunction thesecorrespond to the canonical morphisms

F × p∗Spc(G)→ p∗SpcpSpc] (F)× p∗Spc(G)

∼−→ p∗Spc(pSpc] (F)× G)

induced by the counit of the adjunction (pSpc] , p∗Spc) and the monoidality of the functor p∗Spc.

40 ADEEL A. KHAN

By right transposition we also obtain two canonical morphisms

(6.9) p∗SpcHom(F,G)→ Hom(p∗SpcF, p∗SpcG)

for SmE∞ -fibred spaces F,G ∈ SpcE∞(S) and

(6.10) Hom(pSpc] F,G)→ pSpc

∗ Hom(F, p∗SpcG)

for SmE∞ -fibred spaces F ∈ SpcE∞(X) and G ∈ SpcE∞(S).

The following verifies the left projection formula for SpcE∞ along smooth morphisms, in thesense of Paragraph A.4:

Proposition 6.3.1.

(i) The canonical morphism (6.8) is invertible for all SmE∞-fibred spaces F over X and G over S.

(ii) The canonical morphism (6.9) is invertible for all SmE∞-fibred spaces (resp. motivic spaces)F and G over S.

(iii) The canonical morphism (6.10) is invertible for all SmE∞-fibred spaces F over X and G

over S.

Proof. The claims (ii) and (iii) follow by adjunction from (i). To show that the canonicalmorphism (6.8) is invertible, we may reduce to the case where the spaces F and G are representable,since the functions involved commute with colimits; in this case the claim is clear.

As recalled in Paragraph A.5, the symmetric monoidal functor p∗Spc endows SpcE∞(X) with a

structure of SpcE∞(S)-module category, and the statement of Proposition 6.3.1 is equivalent tothe following:

Corollary 6.3.2. The functor pSpc] lifts to a morphism of SpcE∞(S)-module categories.

6.3.3. Next we verify the corresponding statement at the level of motivic spaces:

Proposition 6.3.4.

(i) The canonical morphism

(6.11) pH] (F × p∗H(G))→ pH

] (F)× G

is invertible for all motivic spaces F over X and G over S.

(ii) The canonical morphism

(6.12) p∗HHom(F,G)→ Hom(p∗HF, p∗HG)

is invertible for all motivic spaces F and G over S.

(iii) The canonical morphism

(6.13) Hom(pH] F,G)→ pH

∗ Hom(F, p∗HG)

is invertible for all motivic spaces F over X and G over S.

Proof. As above, we only need to show claim (i), and only for F and G which are motiviclocalizations of representable spaces; in this case the claim is clear using the formula (6.2) andthe fact that motivic localization commutes with finite products (Lemma 4.3.5).

As above, we also have the following reformulation:

Corollary 6.3.5. The functor pH] lifts to a morphism of HE∞(S)-module categories.


6.3.6. The following slightly more general formula, proved in exactly the same way, will alsobe useful:

Lemma 6.3.7. Let p : X → S be a smooth morphism. Let F be a SmE∞-fibred space (resp.

motivic space) over X, and G→ G′ a morphism of SmE∞-fibred spaces (resp. motivic spaces)over S. Then there is a canonical isomorphism

(6.14) p](F ×p∗(G′)

p∗(G))∼−→ p](F)×

G′G

of SmE∞-fibred spaces (resp. motivic spaces) over S.

6.3.8. At the level of pointed motivic spaces, we get:

Proposition 6.3.9.


(6.15) pH•] (F ∧X p

∗H•(G))→ pH•

] (F) ∧S G

is invertible for all pointed motivic spaces F over X and G over S.


(6.16) p∗H•Hom(F,G)→ Hom(p∗H•F, p∗H•G)

is invertible for all pointed motivic spaces F and G over S.


(6.17) Hom(pH•] F,G)→ pH•

∗ Hom(F, p∗H•G)

is invertible for all pointed motivic spaces F over X and G over S.

Proof. This follows from Lemma 3.3.5 and the smooth projection formula for unpointed spaces(Proposition 6.3.4).

As above, we have the equivalent reformulation:

Corollary 6.3.10. The functor pH•] lifts to a morphism of HE∞(S)•-module categories.

6.3.11. At the level of motivic spectra, we get:

Proposition 6.3.12.


(6.18) pSH] (E ⊗X p

∗SH(E′))→ pSH

] (E)⊗S E′

is invertible for all motivic spectra E over X and G over S.


(6.19) p∗SHHom(E,E′)→ Hom(p∗SHE, p∗SHE

′)

is invertible for all motivic spectra E and E′ over S.


(6.20) Hom(pSH] E,E

′)→ pSH∗ Hom(E, p∗SHE

′)

is invertible for all motivic spectra E over X and E′ over S.

Proof. This follows from Lemma 3.4.6 and the smooth projection formula for pointed motivicspaces (Proposition 6.3.9).

As above, we have the equivalent reformulation:

42 ADEEL A. KHAN

Corollary 6.3.13. The functor pSH] lifts to a morphism of SH(S)S1-module categories.

7. Functoriality along closed immersions

In this section we study some properties of the functor i∗ of direct image along a closedimmersion i. In particular it turns out to admit a right adjoint i!. We also state the localizationtheorem and deduce some of its interesting consequences, which include base change andprojection formulas involving i!.

7.1. The exceptional inverse image functor i!.

7.1.1. Let i : Z → S be a closed immersion. Note that if the base change functor SmE∞/S → SmE∞

/Z

were topologically cocontinuous (see Paragraph B.2), then the direct image functor i∗ on Nisnevichsheaves would commute with arbitrary small colimits. Though this is not quite true, it turnsout that the weaker condition of quasi-cocontinuity (see loc. cit.) holds, which implies that i∗commutes with contractible colimits:

Theorem 7.1.2. Let i : Z → S be a closed immersion. Then the direct image functor iH∗commutes with contractible colimits.

Proof. By Lemma B.2.6 it suffices to show that the base change functor SmE∞/S → SmE∞

/Z is

topologically quasi-cocontinuous in the sense of Lemma B.2.2. This amounts to the following:

(∗) For any smooth spectral S-scheme X and any Nisnevich covering sieve R′ of XZ, thesieve R of X generated by morphisms X′ → X such that either (i) the empty sieve on X′Z isNisnevich-covering, or (ii) X′Z → XZ factors through R′, is Nisnevich-covering.

This condition follows directly from Proposition 2.12.2, which says that etale morphisms canbe lifted (Zariski-locally) along i.

At the level of pointed spaces or spectra, we get:

Corollary 7.1.3. The direct image functor iH•∗ (resp. iSH∗ ) commutes with small colimits.

Proof. It suffices to show that i∗ commutes with contractible colimits and preserves the initialobject. Indeed, given any diagram indexed on an ∞-category I, the initial object defines a coneover this diagram, which is a new diagram that is indexed on a contractible ∞-category andwhich evidently has the same colimit.

On pointed spaces or spectra, it is obvious that i∗ preserves the initial object (= terminalobject). Hence it suffices to note that i∗ commutes with contractible colimits, which followsdirectly from the unpointed statement (Theorem 7.1.2).

By the adjoint functor theorem we obtain a right adjoint i!H• (resp. i!SH), called the exceptionalinverse image functor.

7.2. The localization theorem. In this paragraph, we formulate the statement of the local-ization theorem and explore several of its consequences. The proof will be deferred to the nextsection.

Throughout the paragraph we will work in the category of motivic spaces, and will omit thedecoration HE∞ from the notation for simplicity.


7.2.1. Let i : Z → S be a closed immersion of spectral schemes with quasi-compact opencomplement j : U → S. We deduce some immediate consequences of smooth base change in thissituation.

Considering the commutative square

U U

U S,

j

j

which is cartesian because j is a monomorphism, we get:

Lemma 7.2.2. For any quasi-compact open immersion j : U → S, the canonical morphisms

id→ j∗j],(7.1)

j∗j∗ → id,(7.2)

are invertible.

In other words, the functors j] and j∗ are fully faithful.

7.2.3. Considering the cartesian square

∅ Z

U S,

i

j

we get:

Lemma 7.2.4. For any closed immersion i : Z → S with quasi-compact open complementj : U → S, the canonical morphisms

∅Z → i∗j](FU),

j∗i∗(FZ)→ ptU,

are invertible, for FU (resp. FZ) a motivic space over U (resp. Z).

7.2.5. Consider the canonical commutative square

j]j∗(F) F

j]j∗i∗i

∗(F) i∗i∗(F)

for any motivic space F over S.

By Lemma 7.2.4 this induces a canonical commutative square

(7.3)

j]j∗(F) F

j](ptU) i∗i∗(F)

which we call the localization square associated to the pair (i, j).

The main result in this paper is the following, due to [MV99] in the setting of classicalalgebraic geometry:

Theorem 7.2.6 (Localization). Let i : Z → S be a closed immersion of spectral schemeswith quasi-compact open complement j : U → S. Then for every motivic space F over S, thelocalization square (7.3) is cocartesian.

44 ADEEL A. KHAN

The proof will occupy Sect. 8.

7.2.7. We can deduce from Theorem 7.2.6 a pointed version:

Corollary 7.2.8. Let i : Z → S be a closed immersion of spectral schemes with quasi-compactopen complement j : U → S. For any pointed motivic space (F, s) over S, there is a canonicalcofibre sequence

(7.4) j]j∗(F, s)→ (F, s)→ i∗i

∗(F, s).

and dually, a canonical fibre sequence

(7.5) i∗i!(F, s)→ (F, s)→ j∗j

∗(F, s)

of motivic spaces over S.

Proof. We want to show that the commutative square of pointed motivic spaces

j]j∗(F, x) (F, x)

ptSpc•S i∗i

∗(F, x)

is cocartesian.

Since the forgetful functor (F, x) 7→ F reflects contractible colimits (Lemma 3.3.4), it sufficesto show that the induced square of underlying motivic spaces

j]j∗Ftj]j∗(ptS) ptS F

ptHS i∗i

∗F.

is cocartesian.

Consider the composite square

j]j∗F (j]j

∗F)tj]j∗(ptS) ptS F

j]j∗(ptS) ptS i∗i

∗F.

which is cocartesian by Theorem 7.2.6.

Since the left-hand square is evidently cocartesian, it follows that the right-hand square isalso cocartesian.

7.2.9. Similarly we also deduce localization for motivic S1-spectra:

Corollary 7.2.10. Let i : Z → S be a closed immersion, with quasi-compact open complementj : U → S. For any motivic S1-spectrum E over S, there are canonical exact triangles

j]j∗(E)→ E→ i∗i

∗(E),(7.6)

i∗i!(E)→ E→ j∗j

∗(E),(7.7)

of motivic S1-spectra over S.

Proof. It suffices to show the first sequence is a cofibre sequence. Since the functors in questioncommute with small colimits, Lemma 4.5.3 allows us to the reduce to the case of pointed motivicspaces, which is Corollary 7.2.8.


7.3. A reformulation of localization.

7.3.1. An immediate corollary of Theorem 7.2.6 is:

Corollary 7.3.2. Let i : Z → S be a closed immersion of spectral schemes with quasi-compactopen complement. Then the direct image functor iH∗ (resp. iH•∗ , iSH

∗ ) is fully faithful, and itsessential image is spanned by motivic spaces F for which the restriction j∗H(F) (resp. j∗H•(F),j∗SH(F)) is contractible.

Proof. The claims for iH•∗ and iSH∗ follow directly from that of iH∗ .

First we show that iH∗ is fully faithful. Considering the localization square for iH∗ (F), we seethat the co-unit morphism iH∗ i

∗Hi

H∗ → iH∗ is invertible. By a standard argument it therefore

suffices to show that iH∗ is conservative.

For this, let ϕ : F1 → F2 be a morphism of motivic spaces over Z such that iH∗ (ϕ) is invertible.To show that ϕ is invertible, it suffices to show that

Γ(X,F1)→ Γ(X,F2)

is invertible for each smooth spectral Z-scheme X.

By Proposition 2.12.2, we may assume that X is the base change of a smooth spectral S-schemeY. In this case the claim follows by assumption, since Γ(X,Fk) ≈ Γ(Y, iH∗ (Fk)) for each k, byadjunction.

Next we consider the claim about the essential image. We already know from Lemma 7.2.4that, for any object F in the essential image, the restriction j∗H(F) is contractible. Conversely,given a motivic space F over S, as soon as j∗H(F) is contractible, we obtain from the localizationsquare that the canonical morphism

F → iH∗ i∗H(F)

is invertible.

7.4. Nilpotent invariance. A particularly important consequence of the localization theoremwill be formulated in this paragraph.

7.4.1. Let S be a spectral scheme. The canonical morphism i : Scl → S is a closed immersionwith empty complement j : ∅ → S. In this situation, the localization theorem immediatelygives:

Corollary 7.4.2 (Nilpotent invariance). Let S be a spectral scheme. Then the adjunctions

i∗ : HE∞(S) HE∞(Scl) : i∗,

i∗ : HE∞(S)• HE∞(Scl)• : i∗,

i∗ : SHE∞(S)S1 SHE∞(Scl)S1 : i∗

are equivalences of categories.

Proof. This follows immediately from Corollary 7.3.2.

Note that the same argument also applies to the closed immersion (Scl)red → Scl, so we alsohave invariance with respect to classical nilpotents.

7.5. Closed base change formula.

46 ADEEL A. KHAN

7.5.1. Consider a cartesian square

(7.8)

XZ X

Z S,

k

g f

i

of spectral schemes, with i and k closed immersions with quasi-compact open complements.

At the level of motivic spaces, there is a canonical 2-morphism

(7.9) f∗i∗ → k∗g∗

constructed in Paragraph A.5.

The following says that HE∞ satisfies the right base change property along closed immersions(see loc. cit.):

Corollary 7.5.2. The 2-morphism (7.9) is invertible at the level of motivic spaces.

Proof. By Corollary 7.3.2 it suffices to show that the 2-morphism

f∗i∗i∗ → k∗g

∗i∗

is invertible. This follows by considering the localization squares associated to the closedimmersions i and k, respectively, and using the smooth base change formula (Proposition 6.2.2).

7.5.3. In the pointed setting, the functor i∗ admits a right adjoint i! (Corollary 7.1.3), so weobtain another 2-morphism by right transposition from (7.9). Hence we have:

Corollary 7.5.4. Given a cartesian square of the form (7.8), the canonical 2-morphisms

k∗g∗ → f∗i∗(7.10)

i!f∗ → g∗k!(7.11)

are invertible at the level of pointed motivic spaces.

7.5.5. At the level of spectra, we have:

Corollary 7.5.6. Given a cartesian square of the form (7.8), the canonical 2-morphisms

k∗g∗ → f∗i∗(7.12)

i!f∗ → g∗k!(7.13)

are invertible at the level of motivic spectra.

7.6. Closed projection formula. Let i : Z → S be a closed immersion with quasi-compactopen complement.

As recalled in Definition A.5.8, there are canonical morphisms

(7.14) i∗(F)× G→ i∗(F × i∗(G))

for any pair of motivic spaces F ∈ HE∞(Z) and G ∈ HE∞(S).

The following verifies the right projection formula for HE∞ along closed immersions, in thesense of Paragraph A.5:

Corollary 7.6.1. The canonical morphism (7.14) is invertible, for all motivic spaces F ∈HE∞(Z) and G ∈ HE∞(S).

Proof. This follows from the localization theorem (Theorem 7.2.6) and the smooth projectionformula (Proposition 6.3.4).


As recalled in Paragraph A.5, the symmetric monoidal functor i∗H endows HE∞(Z) with astructure of HE∞(S)-module category, and the statement of Corollary 7.6.1 is equivalent to thefollowing:

Corollary 7.6.2. The functor iH∗ lifts to a morphism of HE∞(S)-module categories.

7.6.3. In the pointed setting, the functor i∗ admits a right adjoint i! (Corollary 7.1.3), so weobtain a dual form of the isomorphism (7.14) by right transposition:

(7.15) i!HomS(G,F)→ HomZ(i∗G, i!F)

for any pair of pointed motivic spaces F and G over S.

Just as above, we obtain:

Corollary 7.6.4. The canonical morphism (7.14) is invertible, for all pointed motivic spacesF ∈ HE∞(Z)• and G ∈ HE∞(S)•. Dually, the canonical morphism (7.15) is invertible, for allpointed motivic spaces F,G ∈ HE∞(S)•.

As above, this is equivalent to:

Corollary 7.6.5. The functor iH•∗ lifts to a morphism of HE∞(S)•-module categories.

7.6.6. At the level of motivic spectra, we get:

Corollary 7.6.7. The canonical morphism (7.14) is invertible, for all motivic spectra F ∈SH(Z)S1 and G ∈ SHE∞(S)S1 . Dually, the canonical morphism (7.15) is invertible, for all

motivic spectra F,G ∈ SHE∞(S)S1 .

As above, this is equivalent to:

Corollary 7.6.8. The functor iSH∗ lifts to a morphism of SHE∞(S)S1-module categories.

7.7. Smooth-closed base change formula.

7.7.1. Consider a cartesian square of spectral schemes

(7.16)

XZ X

Z S,

k

q p

i

where i and j are closed immersions with quasi-compact open complements, and p and q aresmooth.

There is a canonical 2-morphism

(7.17) p]k∗ → i∗q]

at the level of motivic spaces, constructed in Paragraph A.6.

The following verifies the bidirectional base change property for HE∞ with respect to smoothmorphisms and closed immersions:

Corollary 7.7.2 (Smooth-closed base change). Given a cartesian square of the form (7.16),the 2-morphism (7.17) is invertible at the level of motivic spaces.

Proof. Since the direct image functor k∗ is fully faithful (Corollary 7.3.2), it suffices to showthat the transformation

p]k∗k∗ → i∗q]k

∗,

obtained by pre-composition with k∗, is invertible. This follows directly from the localizationtheorem (Theorem 7.2.6) and the smooth base change formula (Proposition 6.2.2).

48 ADEEL A. KHAN

7.7.3. In the pointed setting, we have canonical 2-morphisms

p]k∗ → i∗q](7.18)

q∗i! → k!p∗(7.19)

where the second is obtained by right transposition from the first.

The same argument as above shows:

Corollary 7.7.4 (Smooth-closed base change). Given a cartesian square of the form (7.16),the 2-morphisms (7.18) and (7.19) are invertible at the level of pointed motivic spaces.

7.7.5. At the level of motivic spectra, we have:

Corollary 7.7.6 (Smooth-closed base change). Given a cartesian square of the form (7.16),the 2-morphisms (7.18) and (7.19) are invertible at the level of motivic spectra.

8. The localization theorem

This section is dedicated to the proof of the localization theorem (see Paragraph 7.2).

Throughout the section, we let i : Z → S be a closed immersion of spectral schemes, suchthat the complementary open immersion j : U → S is quasi-compact.

8.1. The space of Z-trivialized maps. In this paragraph we formulate Proposition 8.1.9,which aside from Theorem 7.1.2 is the main input that goes into the proof of the localizationtheorem; most of the remainder of this section will be dedicated to its proof.

Let X be a smooth spectral S-scheme and let t : Z → XZ be a section of the base changeXZ := X×S Z. Proposition 8.1.9 asserts the motivic contractibility of a certain fibred spacehS(X, t); the latter can be thought of as a modified version of the representable space hS(X)whose sections we refer to informally as Z-trivialized morphisms to X.

8.1.1. Given a smooth spectral S-scheme X, let XU := X×S U denote its base change along j,and XZ := X×S Z its base change along i.

We will write hZS(X) for the space over S defined by the cocartesian square

(8.1)

hS(XU) hS(X)

hS(U) hZS(X).

Note that there is a canonical isomorphism

(8.2) i∗Spc(hZS(X)) ≈ hZ(XZ)

of spaces over Z, since i∗Spc commutes with colimits.

Since colimits in SpcE∞(S) are computed section-wise, we can describe the spaces of sections

of hZS(X) explicitly:

Lemma 8.1.2. Let Y be a smooth spectral S-scheme. If YZ is the empty spectral scheme, thenthe space Γ(Y,hZ

S(X)) is contractible. Otherwise, there is a canonical isomorphism of spaces

(8.3) Γ(Y,hZS(X)) ≈ Γ(Y,hS(X)) ≈ MapsS(Y,X).


8.1.3. Let p : X→ S be a smooth morphism. Let t : Z → X be an S-morphism, i.e. a partiallydefined section of p.

Consider the canonical morphism

(8.4) ε : hZS(X)→ iSpc

∗ i∗Spc(hZS(X)) ≈ iSpc

∗ (hZ(XZ))

induced by the counit of the adjunction (i∗Spc, iSpc∗ ).

The morphism t corresponds by adjunction to a morphism τ : hS(S) → iSpc∗ (hS(XZ)). We

define a space hS(X, t) over S as the fibre of ε at the point τ , so that we have a cartesian square

hS(X, t) hZS(X)

hS(S) iSpc∗ (hS(XZ))

ε

τ

of spaces over S.

8.1.4. Sections of the fibred space hS(X, t) can be described as follows, using the fact that

limits in SpcE∞(S) are computed section-wise:

Lemma 8.1.5. Let Y be a smooth spectral S-scheme. If YZ is the empty spectral scheme, thenthe space Γ(Y,hS(X, t)) is contractible. Otherwise, Γ(Y,hS(X, t)) is canonically identified withthe fibre of the restriction map

MapsS(Y,X)→ MapsZ(YZ,XZ)

at the point defined by the composite YZ → Zt→ XZ.

In other words, points of the space Γ(Y,hS(X, t)) (when YZ 6= ∅) are pairs (f, α), withf : Y → X an S-morphism and α a commutative triangle

(8.5)

YZ XZ.

Z

fZ

t

We refer to such a pair (f, α) informally as a Z-trivialized morphism from Y to X.

8.1.6. If p is a smooth morphism, then since p∗Spc commutes with both limits and colimits, wehave:

Lemma 8.1.7. Let X be a smooth spectral S-scheme and t : Z → X an S-morphism. If p : T→ Sis a smooth morphism, then there is a canonical isomorphism of fibred spaces

p∗Spc(hS(X, t)) ≈ hT(XT, tT),

where tT : ZT → XT is obtained from t by base change along p.

8.1.8. Our main result about the fibred space hS(X, t) is as follows:

Proposition 8.1.9. Let p : X→ S be a smooth morphism of affine spectral schemes. Then forevery S-morphism t : Z → X, the space hS(X, t) is motivically contractible, i.e. the morphismhS(X, t)→ ptS is a motivic equivalence.

The proof will occupy the rest of this section.

50 ADEEL A. KHAN

8.1.10. We first consider the case of vector bundles:

Lemma 8.1.11. Let E be a vector bundle over S with zero section s : S → E. Then the spacehS(E, sZ) is motivically contractible, where sZ : Z → EZ denotes the base change of s alongi : Z → S.

Proof. It suffices to construct a section σ of the morphism

ϕ : hS(E, sZ)→ hS(S),

and provide an A1-homotopy between the composite σ ϕ and the identity.

The section

σ : hS(S)→ hS(E, sZ)

is induced by the composite hS(S)s−→ hS(E)→ hZ

S(E).

It remains to define the A1-homotopy

ϑ : hS(A1S)× hS(E, sZ)→ hS(E, sZ).

For each smooth spectral S-scheme Y with YZ 6= ∅, the map

Γ(Y, ϑ) : Γ(Y,hS(A1S))× Γ(Y,hS(E, sZ))→ Γ(Y,hS(E, sZ))

is given by the assignment

(a : Y → A1S, f : Y → E) 7→ (a · f : Y → E).

8.2. Etale base change. In this paragraph we show that the space hS(X, t) is invariant underetale base change.

8.2.1. The assignment (X, t) 7→ hS(X, t) is functorial in the following sense.

Let (X, t) and (X′, t′) be pairs, with X (resp. X′) a smooth spectral S-scheme, and t : Z → X(resp. t′ : Z → X′) a partially defined section. Suppose f : X′ → X is an S-morphism such thatthe square

Z X′Z

Z XZ

t′

t

is cartesian. Then there is a canonical morphism of spaces over S

(8.6) hS(X′, t′)→ hS(X, t).

8.2.2. Let (X, t) and (X′, t′) be as above. Suppose that p : X′ → X is an etale morphism, suchthat the above square is cartesian. Then we have:

Lemma 8.2.3. The canonical morphism

hS(X′, t′)→ hS(X, t)

is a Nisnevich-local equivalence.

Let ϕ denote the morphism in question. The claim is that the induced morphism of Nisnevichsheaves LNis(ϕ) is invertible. It suffices to show that it is 0-truncated (i.e. its diagonal is amonomorphism) and 0-connected (i.e. it is an effective epimorphism and so is its diagonal).


8.2.4. Proof of Lemma 8.2.3, step 1. To show that LNis(ϕ) is 0-truncated, it suffices to showthat ϕ is 0-truncated (since LNis is exact). For this, it suffices to show that for each smoothspectral S-scheme Y, the induced morphism of spaces of Y-sections

Γ(Y, ϕ) : Γ(Y,hZS(X′, t′))→ Γ(Y,hZ

S(X, t))

is 0-truncated.

We may assume YZ is not empty; then this is the morphism induced on fibres in the diagram

Γ(Y,hZS(X′, t′)) MapsS(Y,X′) MapsZ(YZ,X

′Z)

Γ(Y,hZS(X, t)) MapsS(Y,X) MapsZ(YZ,XZ)

Note that the two right-hand vertical morphisms are 0-truncated: p is itself 0-truncated sinceit is etale, and since the Yoneda embedding commutes with limits, the induced morphismhS(X′) → hS(X) is also 0-truncated. It follows that the left-hand vertical morphism is also0-truncated for each Y, and therefore so is ϕ.

8.2.5. Proof of Lemma 8.2.3, step 2. To show that LNis(ϕ) is an effective epimorphism, it sufficesto show that for each smooth spectral S-scheme Y (with YZ not empty), any Y-section of

hZS(X, t) can be lifted Nisnevich-locally along ϕ.

Let f be a Y-section of hZS(X, t), i.e. a Z-trivialized morphism f : Y → X. Let q : Y′ → Y

denote the base change of p : X′ → X along f :

Y′ Y

X′ X.

q

g f

p

Then note that

q−1(YU) Y′

YU Y

q

is a Nisnevich square. Indeed, the closed immersion YZ → Y is complementary to YU → Y,and it is clear that q−1(YZ)→ YZ is invertible because in the diagram

q−1(YZ) YZ

Z Z

X′Z XZ

idZ

t t′

pZ

the lower square and the composite square are cartesian, and hence so is the upper square.

Hence it suffices to show that the restriction of f to either component of this Nisnevichcover lifts to hZ

S(X′, t′). The restriction f |Y′ lifts to a section of hZS(X′, t′) given by g : Y′ → X′.

The restriction f |YU admits a lift trivially: since (YU)×S Z = ∅, the spaces hZS(X, t)(YU) and

hZS(X′, t′)(YU) are both contractible.

52 ADEEL A. KHAN

8.2.6. Proof of Lemma 8.2.3, step 3. It remains to show that the diagonal ∆LNis(ϕ) of LNis(ϕ) isan effective epimorphism, or equivalently that LNis(∆ϕ) is.

For each smooth spectral S-scheme Y, the diagonal induces a morphism of spaces

Γ(Y,hZS(X′, t′))→ Γ(Y,hZ

S(X′, t′)) ×Γ(Y,hZ

S(X,t))Γ(Y,hZ

S(X′, t′)).

It suffices to show that for each Y (with YZ not empty), any point of the target lifts Nisnevich-locally to a point of the source. Choose a point of the target, given by two Z-trivializedmorphisms f : Y → X′ and g : Y → X′, and an identification α : p f ≈ p g.

Let Y0 → Y denote the open immersion defined as the equalizer of the pair (f, g); note thatthe closed immersion YZ → Y factors through Y0. Thus the open immersions Y0 → Y andYU → Y form a Zariski cover of Y. It is clear that the point (f, g, α) lifts after restriction to Y0

by definition, and after restriction to YU since YU×S Z = ∅, so the claim follows.

8.3. Reduction to the case of vector bundles. In this paragraph we show that, Nisnevich-locally, any partially defined section of a smooth spectral S-scheme can be lifted to a globallydefined section.

Together with the etale base change property demonstrated in the last paragraph, andLemma 2.13.9, this will allow us to to reduce to the case of vector bundles.

8.3.1. The following lemma will allow us to reduce to the situation where the Z-section t liftsto an S-section s : S → X.

Lemma 8.3.2. Let p : X→ S be a smooth morphism. Given an S-morphism t : Z → X, thereexists a Nisnevich square

(8.7)

YU Y

U S

q

j

such that q factors through p.

Proof. We will construct a commutative square

(8.8)

Z Y

XZ X.

with the following properties:

(i) The induced square of underlying classical schemes

Zcl Ycl

(XZ)cl Xcl

is cartesian.

(ii) The composite morphism Y → X→ S is etale.

Given such a square (8.8), it is clear that we get a Nisnevich square (8.7) as claimed, by takingq to be the composite Y → X→ S: indeed, the closed immersion Zcl → S is complementary to


j, and the squares

Zcl Ycl Y

Zcl Scl S

q

are cartesian: the left-hand one by (i), and the right-hand one by flatness of q, which is impliedby (ii).

In the classical case, the existence of the square (8.8) is known (this is a non-equivariantversion of [Hoy17, Thm. 2.21], for instance).

Hence one obtains a cartesian square

Zcl Y0

(XZ)cl Xcl

of classical schemes. Then one defines Y by the cocartesian square of closed immersions

Zcl Y0

Z Y.

By Lemma 2.11.2 this exists, and the morphism Y0 → Y is a closed immersion identifying Y0

with the classical scheme underlying Y. The existence of the desired commutative square

Z Y

XZ X

follows by construction.

8.4. Motivic contractibility of hS(X, t). In this paragraph we prove Proposition 8.1.9.

Let p : X → S be a smooth morphism of affine spectral schemes. Recall the statement ofProposition 8.1.9: we want to show that for any S-morphism t : Z → X, the fibred space hS(X, t)is motivically contractible.

8.4.1. By Lemma 8.3.2 there exists a Nisnevich square

(8.9)

YU Y

U S

q

j

where q factors through p : X → S. It suffices then by the Nisnevich separation property(Proposition 6.1.6) to show that j∗ hS(X, t) and q∗ hS(X, t) ≈ hY(Y×S X, t′) are contractible,where t′ : YZ → (Y×S X)Z is the base change of t.

8.4.2. The case of j∗ hS(X, t) is clear, since j is complementary to i : Z → S.

54 ADEEL A. KHAN

8.4.3. For q∗ hS(X, t), note that by construction there exists a section t′′ : Y → Y×S X whichlifts t′ (since q factors through X):

(Y×S X)Z Y×S X

YZ Y

t′ t′′

Hence by Lemma 2.13.9, Lemma 8.2.3 and Lemma 8.1.11, we have motivic equivalences

hS(Y×S

X, t′) ≈ hS(N∗t′′ , z) ≈ hS(S),

where N∗t′′ is the conormal bundle, and z is the base change of its zero section.

8.5. Proof of the localization theorem. We conclude this section by proving the localizationtheorem.

Recall that our goal is to show that the canonical morphism

(8.10) F tj]j∗(F)

MS(U)→ i∗i∗(F)

is invertible for each motivic space F over S.

After a series of reductions, we will see that the main ingredient in the proof is Proposition 8.1.9(which was proved in Paragraph 8.4).

8.5.1. First of all, note that we may use Zariski separation (Proposition 6.1.6) and the smoothbase change formula (Proposition 6.2.2) to reduce to the case where S is affine.

Next note that we may reduce to the case where F is a motivic localization MS(X) of anaffine smooth spectral S-scheme X. Indeed, we have seen that the category HE∞(S) is generatedunder sifted colimits by such objects (Lemma 4.3.5) and that each of the functors j], j

∗, i∗, andi∗ commutes with contractible colimits (Theorem 7.1.2).

In this case the morphism (8.10) is canonically identified with the morphism

(8.11) MS(X) tMS(XU)

MS(U)→ i∗MS(XZ)

where we write XU = X×S U and XZ = X×S Z.

8.5.2. Note that the source of the morphism (8.11) is the motivic localization of the space

hZS(X), and that the target iH∗ (MZ(XZ)) is the motivic localization of iSpc

∗ (hZ(XZ)).

Hence it suffices to show that the morphism

(8.12) hZS(X)→ iSpc

∗ hZ(XZ)

is a motivic equivalence.

8.5.3. By universality of colimits, it suffices to show that for every smooth spectral S-scheme Y

and every morphism hS(Y)→ iSpc∗ hZ(XZ), corresponding to an S-morphism t : Z→ X, the base

change

(8.13) hZS(X) ×

iSpc∗ hZ(XZ)

hS(Y)→ hS(Y)

is invertible.


8.5.4. Let p : Y → S be the structural morphism of Y. Then since hS(Y) ≈ pSpc] hY(Y), one

sees that (8.13) is identified, by the smooth projection formula (Lemma 6.3.7), with a morphism

(8.14) pSpc] (p∗Spc hZ

S(X) ×p∗Spci

Spc∗ hZ(XZ)

hY(Y))→ pSpc] hY(Y).

8.5.5. Note that we have p∗i∗ ≈ k∗q∗ (Proposition 6.2.2), where k (resp. q) is the base changeof i (resp. p) along p (resp. i). Hence the morphism (8.14) is identified with the image by p] of

(8.15) hYZ

Y (X×S

Y) ×kSpc∗ hYZ

((X×S Y)Z)

hY(Y)→ hY(Y).

8.5.6. The source of the morphism (8.15) is nothing else than the space hY(X×S Y, tY), wheretY : Z×S Y → X×S Y is the base change of t along p. Hence we conclude by Proposition 8.1.9.

Appendix A. Coefficient systems

In this section we will introduce a formalism for working with categories of coefficients. Theseare systems of categories D(S), indexed by objects in some category C (e.g. schemes), equippedwith some basic functorialities f∗ and f∗ associated to any morphism f in C. This is a naturalframework for stating base change and projection formulas.

We will use the language of (∞, 2)-categories as a convenient way to formulate some of theresults in this section, even though we only apply them in the (∞, 2)-category of presentable∞-categories. Following [GR16], our definition of (∞, 2)-category will be a complete Segal spacein the ∞-category of ∞-categories.

A.1. Passage to right/left adjoints.

A.1.1. Recall that in an (∞, 2)-category C, there is a notion of adjunction between two objectsx and y.

A pair (f : x→ y, g : y → x) forms an adjunction if and only if it defines an adjunction inthe underlying ordinary 2-category (C)2-ordn.

A.1.2. Let C and D be (∞, 2)-categories. We say that an (∞, 2)-functor u : C→ D is right-adjointable (resp. left-adjointable) if, for each 1-morphism f : x→ y in C, its image u(f) admitsa right adjoint (resp. a left adjoint) in the 2-category D.

A.1.3. Let Maps!(S,T) denote the space of right-adjointable (∞, 2)-functors. Let Maps∗(S,T)denote the space of left-adjointable (∞, 2)-functors.

Lemma A.1.4. There is a canonical isomorphism of spaces

Maps!(S,T) = Maps∗((S)1&2-op,T),

where (C)1&2-op denotes the (∞, 2)-category obtained by flipping the directions of 1- and 2-morphisms in S.

See [GR16, Cor. 1.3.4].

Given a right-adjointable (∞, 2)-functor u : S → T, we will call the corresponding functoru∗ : (S)1&2-op → T the functor obtained from u by passage to right adjoints.

Dually, given a left-adjointable (∞, 2)-functor u : S → T, we will call the correspondingfunctor u! : (S)1&2-op → T the functor obtained from u by passage to left adjoints.

56 ADEEL A. KHAN

A.2. Adjointable squares. In this section we will formulate the notion of horizontally/verticallyleft/right-adjointable square in a 2-category, which we will be used in the text to express basechange formulas.

A.2.1. We fix an (∞, 2)-category C.

Let Θ be a square in C

(A.1)

C C′

D D′

u

v v′

u′

which commutes up to an invertible 2-morphism

v′u∼−→ u′v.

Suppose that v (resp. v′) admits a right adjoint vR (resp. (v′)R) in C. Then the squareΘvert:R

(A.2)

C C′

D D′

u

u′

vR (v′)R

commutes up to the 2-morphism

(A.3) uvR → (v′)Rv′uvR ∼−→ (v′)Ru′vvR → (v′)Ru′,

where the first morphism is obtained by precomposition with the counit of the adjunction,the isomorphism in the middle is given by the commutativity of the square Θ, and the finalmorphism is given by the unit of the adjunction.

If this 2-morphism is invertible, then we say that the square Θ is vertically right-adjointable.

A.2.2. Similarly if v (resp. v′) admits a left adjoint vL (resp. (v′)L), then the square Θvert:L

(A.4)

C C′

D D′

u

u′

vL (v′)L

commutes up to the 2-morphism

(A.5) (v′)Lu′ → (v′)Lu′vvL ∼←− (v′)Lv′uvL → uvL.

If this is invertible, we say that the square Θ is vertically left-adjointable.

A.2.3. If u (resp. u′) admits a right adjoint uR (resp. (u′)R), then the square Θhoriz:R

(A.6)C C′

D D

v v′uR

(u′)R

commutes up to a 2-morphism

(A.7) vuR → (u′)Rv′.

If it is invertible, we say that Θ is horizontally right-adjointable.


Similarly, if u (resp. u′) admits a left adjoint uL (resp. (u′)L), then the square Θhoriz:L

(A.8)C C′

D D

v v′uL

(u′)L

commutes up to a 2-morphism

(A.9) (u′)Lv′ → vuL.

If it is invertible, we say that Θ is horizontally left-adjointable.

A.2.4. We have:

Lemma A.2.5. Suppose that in the square Θ (A.1), u (resp. u′) admits a right adjoint uR

(resp. (u′)R), and v (resp. v′) admits a left adjoint vL (resp. (v′)L). Then Θ is verticallyleft-adjointable if and only if it is horizontally right-adjointable.

Proof. The square Θvert:L (resp. Θhoriz:R) commutes up to a 2-morphism α : (v′)Lu′ → uvL

(resp. β : vuR → (u′)Rv′). The category of left adjoint functors C → D is equivalent to thecategory of right adjoint functors D→ C (see [GR16, Chap. A.3, Cor. 3.1.9]), and under thisequivalence the morphism α corresponds to the morphism β.

A.3. Coefficient systems. Let Pres denote the symmetric monoidal∞-category of presentable∞-categories. Recall that a presentable∞-category is an accessible left localization of a presheafcategory PSh(C) with C essentially small, and a morphism of presentable ∞-categories is acolimit-preserving functor.

For the rest of this section, we fix an essentially small ∞-category C which admits fibredproducts. Note that the cartesian monoidal structure on C induces a canonical cocartesianmonoidal structure on (C)op.

A.3.1.

Definition A.3.2. A coefficient system (defined on C) is a symmetric monoidal functor

D∗ : (C)op → Pres.

Given a coefficient system D∗, we will write

D(S) := D∗(S)

for the presentable ∞-category associated to an object S ∈ C. We will write ∅S (resp. eS) forthe initial (resp. terminal) object of D(S). We will often refer to the objects of D(S) as sheaveson S.

For a morphism f : T→ S, we will write

f∗ := D∗(f) : D(S)→ D(T)

for the induced functor, which we call the functor of inverse image along f . It is cocontinuous,and admits (by the adjoint functor theorem) a right adjoint

f∗ : D(T)→ D(S)

which we call the functor of direct image along f .

58 ADEEL A. KHAN

A.3.3. By passing to right adjoints (see Paragraph A.1), D∗ gives rise to a unique functor

D∗ : C→∞-Cat

such that each functor

D∗(f) : D(T)→ D(S)

is the right adjoint f∗.

A.3.4. Since the functor D∗ underlying a coefficient system is symmetric monoidal, it sendscocommutative comonoids in C to commutative monoids in Pres. Note that every object S ∈ Chas a canonical structure of cocommutative comonoid (with respect to the cartesian monoidalstructure).

Hence for each object S ∈ C, the presentable ∞-category D(S) has a canonical symmetricmonoidal structure, and for each morphism f in C, the inverse image functor f∗ has a canonicalsymmetric monoidal structure (giving by adjunction a lax monoidal structure on its right adjointf∗).

We will write ⊗S for the monoidal product of D(S), 1S for the monoidal unit, and HomS forthe internal hom.

A.3.5. Dually, suppose we are given a functor (not necessarily symmetric monoidal)

D! : C→ Pres.

We will write D(S) := D!(S) for the presentable ∞-category associated to an object S ∈ C. Foreach morphism f : T→ S in C, we write

f! := D!(f) : D(T)→ D(S)

for the induced functor, which commutes with colimits and admits a right adjoint f !.

As above, we can pass to right adjoints to obtain a functor D! : (C)op → Pres.

A.3.6. For future use, we make the following definition:

Definition A.3.7. A coefficient system D∗ is pointed (resp. S1-stable, compactly generated)if the functor D∗ : (C)op → Pres factors through the full subcategory spanned by pointed (resp.stable, compactly generated) presentable ∞-categories.

A.4. Left-adjointability.

A.4.1. Let us fix a class left of left-admissible morphisms in C, containing all isomorphisms,closed under composition and base change, and satisfying the 2-out-of-3 property. Let Cleft

denote the (non-full) subcategory of C spanned by left-admissible morphisms.

Definition A.4.2. We say that the coefficient system D∗ is weakly left-adjointable along amorphism p : T→ S if it satisfies the following property:

(Adjp) The functor p∗ admits a left adjoint p].

We say that D∗ is weakly left-adjointable along the class left if it satisfies the followingproperty:

(Adjleft) For every left-admissible morphism p, the property (Adjp) holds.

Note that if D∗ is weakly left-adjointable along left , then one obtains a canonical functor

D] : Cleft → Pres

by passage to left adjoints (see Paragraph A.1).


A.4.3. Recall the notion of adjointability of squares from Paragraph A.2.

Definition A.4.4. The coefficient system D∗ satisfies left base change along a morphismp : S′ → S if it is weakly left-adjointable along left, and the following property holds:

(BCp) For all cartesian squares Θ

(A.10)

T′ S′

T S,

f ′

p′ p

f

the induced commutative square Θ∗

(A.11)

D(S) D(T)

D(S′) D(T′).

f∗

p∗ (p′)∗

(f ′)∗

is vertically left-adjointable.

We say that D∗ satisfies left base change along the class left if it is weakly left-adjointablealong left, and the following property holds:

(BCleft) For every left-admissible morphism p, the property (BCp) holds.

In other words, D∗ satisfies left base change along a morphism p if for every such cartesiansquare Θ, the exchange 2-morphism

(A.12) (p′)](f′)∗ → f∗p]

is invertible.

According to Lemma A.2.5, this is equivalent to the condition that its right transpose

(A.13) p∗f∗ → (f ′)∗(p′)∗

is invertible.

A.4.5. For any morphism f : T→ S, the symmetric monoidal functor f∗ : D(S)→ D(T) givesD(T) a structure of D(S)-module category. If f] is left adjoint to f∗, then it admits a canonicalstructure of colax morphism of D(S)-modules. In particular there are canonical morphisms

f](F ⊗T f∗(G))→ f](F)⊗S G (F ∈ D(T),G ∈ D(S)).

Definition A.4.6. The coefficient system D∗ satisfies the left projection formula along amorphism p : T→ S if it is weakly left-adjointable along p, and the following property holds:

(Projp) The colax morphism of D(S)-modules p] is strict.

We say that D∗ satisfies the left projection formula along the class left if it is weaklyleft-adjointable along left, and the following property holds:

(Projleft) For every left-admissible morphism p, the property (Projleft) holds.

In other words, D∗ satisfies the left projection formula along p : T → S if the canonicalmorphisms

(A.14) p](F ⊗T p∗(G))→ p](F)⊗S G (F ∈ D(T),G ∈ D(S))

are invertible.

60 ADEEL A. KHAN

A.4.7.

Definition A.4.8. A coefficient system D∗ is left-adjointable along the class left if it is weaklyleft-adjointable along left (Adjleft ), satisfies left base change along left (BCleft ), and satisfies the

left projection formula along left (Projleft).

A.5. Right-adjointability.

A.5.1. Let us fix a class right of right-admissible morphisms in C, containing all isomorphisms,and closed under composition and base change. Let Cright denote the (non-full) subcategory ofC spanned by right-admissible morphisms.

In this paragraph we consider the dual versions of the definitions given in Paragraph A.4.

A.5.2. Let D∗ be a coefficient system.

Definition A.5.3. The coefficient system D∗ is weakly right-adjointable along a morphism qif it satisfies the following property:

(Adjq) The direct image functor q∗ admits a right adjoint.

We say that D∗ is weakly right-adjointable along a class right if it satisfies the followingproperty:

(Adjright) For each right-admissible morphism q, the property (Adjq) holds.

Remark A.5.4. Note that the situation is not completely symmetric: by requiring that ourcoefficient system D∗ takes values in Pres, we have imposed (by the adjoint functor theorem)that the inverse image functor f∗ always admits a right adjoint f∗, for arbitrary morphisms f . Inthe above definition of weak right-adjointability, we are instead imposing the condition that theright adjoint f∗ itself admits a further right adjoint f ! (when the morphism f is right-admissible).

A.5.5. We define:

Definition A.5.6. The coefficient system D∗ satisfies right base change along a morphismq : S′ → S if the following property holds:

(BCright) For all cartesian squares Θ in C

T′ S′

T S

f ′

q′ q

f

with q and q′ right-admissible, the induced commutative square Θ∗

D(S) D(T)

D(S′) D(T′).

f∗

q∗ (q′)∗

(f ′)∗

is vertically right-adjointable.

We say that D∗ satisfies right base change along the class right if the following propertyholds:

(BCright) For every right-admissible morphism q, the property (BCq) holds.


In other words, D∗ satisfies right base change along a morphism q if for every such cartesiansquare Θ, the exchange 2-morphism

f∗q∗ → (q′)∗(f′)∗

is invertible.

A.5.7. As already mentioned, for any morphism f : T→ S, the functor f∗ endows D(T) with acanonical structure of D(S)-module category. It follows by adjunction that the right adjoint f∗admits a canonical structure of lax morphism of D(S)-modules. In particular there are canonicalmorphisms

f](F)⊗S G→ f](F ⊗T f∗(G)) (F ∈ D(T),G ∈ D(S)).

We define:

Definition A.5.8. The coefficient system D∗ satisfies the right projection formula along amorphism q : T→ S if the following property holds:

(Projq) The canonical structure of lax morphism of D(S)-modules on q∗ is strict.

We say that D∗ satisfies the right projection formula along the class right if the followingproperty holds:

(Projright) For every right-admissible morphism q, the property (Projq) holds.

In other words, D∗ satisfies the right projection formula along q : T → S if the canonicalmorphisms

(A.15) q∗(F)⊗S G→ q∗(F ⊗T q∗(G)) (F ∈ D(T),G ∈ D(S)).

are invertible.

A.5.9. Finally, we define:

Definition A.5.10. A coefficient system D∗ is right-adjointable along the class right if it isweakly right-adjointable along right (Adjright), satisfies right base change along right (BCright),and satisfies the right projection formula along right (Projright).

A.6. Biadjointability.

A.6.1. Let D∗ be a coefficient system which satisfies left base change along left .

This means that, for every cartesian square Θ in C

(A.16)T′ S′

T S

q′

p′ p

q

with p and p′ left-admissible, the square

(A.17)

D(S) D(T)

D(S′) D(T′).

q∗

(q′)∗

p] (p′)]

commutes up to a canonical invertible 2-morphism.

62 ADEEL A. KHAN

One can then ask whether this resulting square is further horizontally right-adjointable, i.e.whether the square

(A.18)

D(S) D(T)

D(S′) D(T′)

q∗

p] (p′)]

(q′)∗

commutes via the 2-morphism

(A.19) p](q′)∗ → q∗q

∗p](q′)∗ ' q∗(p′)](q′)∗(q′)∗ → q∗(p

′)].

We define:

Definition A.6.2. The coefficient system D∗ satisfies bidirectional base change along thepair (left , right) if it satisfies left base change along left, right base change along right, and thefollowing property holds:

(BCleftright) For all cartesian squares Θ in C of the form (A.16), with p and p′ left-admissible

(resp. q and q′ right-admissible), the square (A.18) commutes.

In other words, we require that for all cartesian squares Θ as above, the 2-morphism

(A.20) p](q′)∗ → q∗(p

′)].

is invertible.

According to Lemma A.2.5, this is equivalent to the condition that the right transpose

(A.21) (p′)∗q! → (q′)!p∗

is invertible.

A.6.3. Finally, we define:

Definition A.6.4. A coefficient system D∗ is (left , right)-biadjointable if it left-adjointablealong left, right-adjointable along right, and satisfies bidirectional base change along (left , right)

(BCleftright).

Appendix B. Topologically quasi-cocontinuous functors

B.1. Reduced presheaves.

B.1.1. Let C be an essentially small ∞-category admitting an initial object ∅C.

Recall from Definition 3.6.4 that a presheaf F is reduced if it satisfies the condition that thespace F(∅C) is contractible.

B.1.2. Let Ered denote the empty pre-excision structure on C, containing no cartesian squares.

Assume that the axiom (EXC0) holds, i.e. every morphism c→ ∅C is invertible. The axioms(EXC1), (EXC2) and (EXC3) are vacuous, so Ered is a topological excision structure. Thecorresponding topology is denoted τred; it is defined by the single covering sieve ∅ → h(∅C),where ∅ is the initial presheaf.

The following lemma is tautological (it can be viewed as a trivial special case of Theorem 3.7.9):

Lemma B.1.3. A presheaf F is reduced if and only if it is Ered-excisive, or if and only if itsatisfies τred-descent.


B.1.4. Let PShred(C) denote the ∞-category of reduced presheaves. It is well-known thatPShred(C) is the ∞-category freely generated by C under contractible colimits.

(Recall that a small ∞-category is contractible if the ∞-groupoid formed by formally addinginverses to all morphisms is (weakly) contractible.)

B.1.5. Let Lred denote the τred-localization functor, the exact left adjoint to the inclusionincred : PShred(C) → PSh(C).

For a presheaf F on C, Lred(F) can be described as the unique reduced presheaf for whichthe space Lred(F)(c) is identified with F(c) whenever c is not initial.

B.2. Topologically quasi-cocontinuous functors. Recall that a functor u between sites istopologically cocontinuous11 if the restriction of presheaves functor u∗ preserves local equivalences.

In this paragraph we introduce a slightly weaker version of this condition, which implies thatu∗ preserves local equivalences between presheaves F satisfying the mild condition that F(∅) iscontractible. This will have the useful consequence that u∗ commutes with contractible colimitsat the level of sheaves.

B.2.1. Let C and D be essentially small ∞-categories, admitting initial objects ∅C and ∅D,respectively. Let EC (resp. ED) be a topological excision structure on C (resp. D), and let τC(resp. τD) denote the associated Grothendieck topology.

Recall that a functor u : C → D is topologically cocontinuous if the following condition issatisfied:

(COC) For every τ ′-covering sieve R′ → h(u(c)), the sieve R → h(c), generated by morphismsc′ → c such that h(u(c′))→ h(u(c)) factors through R′, is τ -covering.

We define:

Definition B.2.2. A functor u : C→ D is topologically quasi-cocontinuous if it satisfies thefollowing condition:

(COC’) For every τ ′-covering sieve R′ → h(u(c)), the sieve R → h(c), generated by morphismsc′ → c such that either h(u(c′))→ h(u(c)) factors through R′ → h(u(c)) or u(c′) is initial, isτ -covering.

B.2.3. Let C and D be as above. We have:

Lemma B.2.4. Let u : C → D be a topologically quasi-cocontinuous functor. Then thecomposite functor

u∗red : PShred(D) → PSh(D)u∗−→ PSh(C)

sends τD-local equivalences to τC-local equivalences.

Proof. Recall that the set of τD-local equivalences in PSh(D) is the strongly saturated closureof the set SED

containing the morphism

∅ → h(∅D)

and the morphisms

cQ : CQ → h(d),

11The term cocontinuous is used in [AGV73]. Nowadays this term is often used for colimit-preserving functors,so we have slightly deviated from the classical terminology in order to avoid confusion.

64 ADEEL A. KHAN

for every object d ∈ D and every square Q ∈ ED over d (see (3.11) for notation). In other words,the set of τD-local equivalences is the closure of SED

under the 2-of-3 property, cobase change,and colimits.

It follows that the set of τD-local equivalences in PShred(D) is the closure of the set of themorphisms Lred(s) for s ∈ SED

, under 2-of-3, cobase change, and contractible colimits.

Note that the functor u∗red commutes with contractible colimits, since u∗ commutes with smallcolimits and the inclusion incred commutes with contractible colimits. This implies that it issufficient to show that u∗red sends every morphism Lred(s), for s ∈ SED

, to a τC-local equivalence.

Note that Lred(∅)→ Lred(h(∅D)) is invertible, and the morphisms

Lred(cQ) : Lred(CQ)→ Lred(h(d))

are canonically identified with cQ : CQ → h(d) for every Q and d. The latter fact follows fromthe fact that representable presheaves are reduced, and Lred is exact. This means that we onlyneed to show that u∗red sends each morphism cQ to a τC-local equivalence.

For each square Q ∈ ED of the form

(B.1)

e′ e

d′ d,

g

f

recall that CQ is identified with the τD-covering sieve R′ → h(d) generated by the familyd′ → d, e→ d.

To show that ϑ : u∗red(R′) → u∗red(h(d)) is a τC-local equivalence, it suffices by universality ofcolimits to show that, for every object c of C and every morphism ϕ : h(c)→ u∗redh(d), the basechange

u∗redR′ ×u∗redh(d)

h(c) → h(c)

is a τC-covering sieve.

Note that u∗red admits a left adjoint

ured! : PSh(C)

u!−→ PSh(D)Lred−−−→ PShred(D)

by construction. The morphism ϕ factors canonically through the unit morphism h(c) →u∗redu

red! h(c) = u∗redh(u(c)) and the canonical morphism u∗redu

red! h(c) = u∗redh(u(c))→ u∗redh(d)

(obtained by adjunction from ϕ). The base change of ϑ by u∗redh(u(c))→ u∗redh(d) is identified,since u∗red commutes with limits, with the canonical morphism

u∗(R′ ×h(d)

h(u(c)))→ u∗redh(u(c)).

Since the sieve R′×h(d) h(u(c)) → h(u(c)) is τD-covering, as the base change of a τD-coveringsieve, the conclusion follows directly from the condition (COC’).

B.2.5. Let u : C→ D be as above. The following lemma is a formal consequence of Lemma B.2.4,and the fact that reduced presheaves are stable by contractible colimits:

Lemma B.2.6. Suppose that u is topologically quasi-cocontinuous. Then the composite functor

ShτD(D)incτD−→ PSh(D)

u∗−→ PSh(C)LτC−−−→ ShτC(C)

commutes with contractible colimits.


Proof. The inclusion incτD factors as

incτD : ShτD(D)incredτD−→ PShred(D)

incred−→ PSh(D),

and its left adjoint LτD factors as

LτD : PSh(D)Lred−−−→ PShred(D)

LredτD−−−→ ShτD(D),

with LredτD left adjoint to incred

τD .

Given a diagram (Fi)i∈I of τD-sheaves indexed by a contractible ∞-category I, consider thecanonical morphism

lim−→i∈I

incredτD (Fi) −→ incred

τD LredτD lim−→

i∈I

incredτD (Fi),

which is clearly a τD-local equivalence (where the colimits are taken in PShred(D)). By applyingu∗red = u∗ incred this induces a morphism

u∗red lim−→i∈I

incredτD (Fi) −→ u∗red incred

τD LredτD lim−→

i∈I

incredτD (Fi),

which is identified with a canonical morphism

lim−→i∈I

u∗ incτD(Fi) −→ u∗ incτC lim−→i∈I

Fi

since the inclusion incred commutes with contractible colimits. By Lemma B.2.4, this is aτC-local equivalence, so the claim follows.

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66 ADEEL A. KHAN

Fakultat fur Mathematik, Universitat Regensburg, 93040 Regensburg, Germany

E-mail address: [email protected]

URL: http://www.preschema.com

mailto:[email protected]

http://www.preschema.com

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